<Text-field style="Heading 1" layout="Heading 1">Starting at (0,0)</Text-field>

The basic functional equation. The series Cm(1/x) is negative in x

K:=1-t*(x+1/x+y+1/y):

eqC:=-K*C(x,y)+1-t/y*Cm(1/x)-t/x*Cm(1/y);

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Expansions

Cmser:=proc(n) local res: res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n): end:

Cser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C(x,y)=Cser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),eqC+C(x,y)),t,n+1)): fi: end:

map(expand,Cser(3));

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All walks in C: this sequence appears in the OEIS as A060898

n:=10:subs(x=1,y=1,Cser(n));

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Walks ending at (0,0). This one is not in the OEIS

coeff(coeff(convert(Cser(n),polynom),x,0),y,0);

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Walks in the quadrant.

eqQ:=-K*Q(x,y)+1-t/y*Qp(x)-t/x*Qp(y);

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Expansions

Qpser:=proc(n) subs(y=0,Qser(n)): end:

Qser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q(x,y)=Qser(n-1),Qp(x)=Qpser(n-1),Qp(y)=subs(x=y,Qpser(n-1)),eqQ+Q(x,y)),t,n+1)): fi: end:

Qser(4);

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<Text-field style="Heading 2" layout="Heading 2">Explicit values for quadrant walks </Text-field>

Checking the expression of the paper

Qconj:=proc(i,j,n) if i+j+n mod 2 <>0 then 0 else (i+1)*(j+1)/(n+1)/(n+2)*binomial(n+2,(n-i-j)/2)*binomial(n+2,(n+i-j+2)/2):fi: end:

seq(Qconj(0,0,2*n),n=0..6);

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N:=6: add(add(add(Qconj(i,j,n)*x^i*y^j*t^n,i=0..n),j=0..n),n=0..N);

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

normal(series(Qser(N)-%,t,N+1));

IiIh

<Text-field style="Heading 2" layout="Heading 2">Reduction to an equation with orbit sum zero</Text-field>

eqorbitQ:=x*y*Q(x,y)-y/x*Q(1/x,y)+1/x/y*Q(1/x,1/y)-x/y*Q(x,1/y)-(x-1/x)*(y-1/y);

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We know four solutions to this equation : first, Q(x,y) itself, then -Q(1/x,y)/x^2 :

factor(subs({Q(x,y)=-1/x^2*Q(1/x,y), Q(1/x, y)=-x^2*Q(x,y), Q(x, 1/y)=-1/x^2*Q(1/x,1/y),Q(1/x, 1/y)=-x^2*Q(x,1/y)}, eqorbitQ)/eqorbitQ);

IiIi

by symmetry, -Q(x,1/y)/y^2 and finally Q(1/x,1/y)/x^2/y^2

factor(subs({Q(x,y)=1/x^2/y^2*Q(1/x,1/y), Q(1/x, y)=x^2/y^2*Q(x,1/y), Q(x, 1/y)=y^2/x^2*Q(1/x,y),Q(1/x, 1/y)=x^2*y^2*Q(x,y)}, eqorbitQ)/eqorbitQ);

IiIi

A linear combination

Aser:=proc(n) series(Cser(n)-(1-2*a)*Qser(n)+a/x^2*subs(x=1/x,Qser(n))+a/y^2*subs(y=1/y,Qser(n)),t,n+1): end:

Checking that A has orbit sum zero, whatever a is

n:=6: factor(series(x*y*Aser(n)-y/x*subs(x=1/x,Aser(n))-x/y*subs(y=1/y,Aser(n))+1/x/y*subs(x=1/x,y=1/y,Aser(n)),t,n+1));

IiIh

A functional equation for A

Cexpr:=A(x,y)+(1-2*a)*Q(x,y)-a/x^2*Q(1/x,y)-a/y^2*Q(x,1/y);

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Cmxexpr:=Am(1/x)-a/x^2*Qp(1/x);

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Cmyexpr:=Am(1/y)-a*Qp(1/y)/y^2;

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eqC;

LCoqJiwmIiIiRiUqJkkidEc2IkYlLCpJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiVGJUYsRiUtSSJDR0YoNiRGKkYtRiVGLEYlRiUqKEYnRiVGLUYsLUkjQ21HRig2I0YrRiVGLCooRidGJUYqRiwtRjQ2I0YuRiVGLA==

eqA:=collect(subs(C(x,y)=Cexpr,Cm(1/x)=Cmxexpr,Cm(1/y)=Cmyexpr,eqC-(1-2*a)*eqQ+a/x^2*subs(x=1/x,eqQ)+a/y^2*subs(y=1/y,eqQ)),[A,Am,Q,Qp],factor);

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expand(op(6,eqA));

LCgqJkkiYUc2IiIiIilJInhHRiUiIiMhIiJGJiomRilGJkYkRiZGJiomRiRGJilJInlHRiVGKUYqRiY=

To get rid of Qp

eqA:=map(factor,subs(a=1/3,eqA));

LCoqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRiZGKEYqRihGKComRiZGKEYsRihGKComRipGKEYsRighIiJGKEYqRjJGLEYyLUkiQUdGJzYkRipGLEYoRigqKEYmRihGLEYyLUkjQW1HRic2IyokRipGMkYoRjIqKEYmRihGKkYyLUY4NiMqJEYsRjJGKEYyKiojRigiIiRGKCwoKihGK0YoRilGKEYuRihGKCokRilGKEYoKiRGLkYoRihGKEYpRjJGLkYyRig=

Aser:=proc(n) normal(series(Cser(n)-1/3*Qser(n)+1/3/x^2*subs(x=1/x,Qser(n))+1/3/y^2*subs(y=1/y,Qser(n)),t,n+1)): end:

Aser(2);

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Am is negative in x

Amser:=proc(n) local res: res:=coeff(convert(Aser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end:

n:=16: normal(series(subs(A(x,y)=Aser(n), Am(1/x)=Amser(n), Am(1/y)=subs(x=y,Amser(n)), eqA),t,n+1));

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What we need to remember from the above sub-section:

Aser:=proc(n) normal(series(Cser(n)-1/3*Qser(n)+1/3/x^2*subs(x=1/x,Qser(n))+1/3/y^2*subs(y=1/y,Qser(n)),t,n+1)): end:

We separate in A the contribution of the three quadrants

M(1/x,y) is the negative part of A(x,y) in x (it is automatically non-negative in y)

Mser:=proc(n) collect(add(1/x*coeff(convert(Aser(n),polynom),x,-i)*x^i,i=1..n+2),t,normal): end:

Mser(4);

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P(x,y) is the non-negative part of A in x and y

Pser:=proc(n) local res: res:=convert(Aser(n),polynom): collect(add(add(coeff(coeff(res,x,i),y,j)*x^i*y^j,i=0..n),j=0..n),t): end:

Pser(3);

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Parity properties

n:=10: normal(series(subs(y=0,x=x*t,t*Mser(n)),t,n+1)); normal(series(subs(x=0,y=x*t,t*Mser(n)),t,n+1));

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Ky9JInRHNiIiIiIiIiMsJiIiKEYlKiZGJkYlSSJ4R0YkRiVGJSIiJSwoKiYiIiRGJSlGKkYmRiVGJSomIiNBRiVGKkYlRiUiI2xGJSIiJywqKiZGK0YlKUYqRi5GJUYlKiYiI1tGJUYvRiVGJSomIiRdI0YlRipGJUYlIiQjcEYlIiIpLCwqJiIiJkYlKUYqRitGJUYlKiYiIykpRiVGNkYlRiUqJiIkbSdGJUYvRiVGJSomIiUhKkhGJUYqRiVGJSIlIip6RiUiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

<Text-field style="Heading 2" layout="Heading 2">Reduction to a quadrant-like problem</Text-field>

eqorbitA:=x*y*A(x,y)-y/x*A(1/x,y)+1/x/y*A(1/x,1/y)-x/y*A(x,1/y);

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AexprPM:=A(x,y)=P(x,y)+M(1/x,y)/x+M(1/y,x)/y;

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expand(subs(AexprPM,subs(x=1/x,AexprPM),subs(y=1/y,x=1/x,AexprPM),subs(y=1/y,AexprPM),eqorbitA));

LDoqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYtSSJNR0YlNiQqJEYkISIiRidGJkYmKiZGJEYmLUYtNiQqJEYnRjBGJEYmRiYqKEYkRjBGJ0YmLUYpRi5GJkYwKiZGJ0YmLUYtRipGJkYwKiZGJEYwLUYtNiRGNEYvRiZGMCooRidGMEYkRjAtRik2JEYvRjRGJkYmKiZGJ0YwLUYtNiRGJEY0RiZGJiomRiRGMC1GLTYkRidGL0YmRiYqKEYkRiZGJ0YwLUYpRkFGJkYwKiZGJ0YwLUYtRj5GJkYwKiZGJEYmLUYtNiRGJ0YkRiZGMA==

Extract the positive part in x and y

x*y*P(x,y)-y*(M(x, y)-M(0,y))-x*(M(y,x)-M(0,x));

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PexprM:= expand(solve(%,P(x,y)));

LCoqJkkieEc2IiEiIi1JIk1HRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1GKDYkRipGJEYrRisqJkYqRiYtRig2JEYvRiRGK0Ym

AexprM:=subs(P(x,y)=PexprM,AexprPM);

Ly1JIkFHNiI2JEkieEdGJUkieUdGJSwuKiZGJyEiIi1JIk1HRiVGJiIiIkYuKiZGJ0YrLUYtNiQiIiFGKEYuRisqJkYoRistRi02JEYoRidGLkYuKiZGKEYrLUYtNiRGMkYnRi5GKyomLUYtNiQqJEYnRitGKEYuRidGK0YuKiYtRi02JCokRihGK0YnRi5GKEYrRi4=

Checking the equivalence with the orbit equation for A

expand(subs(AexprM,subs(x=1/x,AexprM),subs(y=1/y,x=1/x,AexprM),subs(y=1/y,AexprM),eqorbitA));

IiIh

Let us check with the series

n:=3:Pser(n);

collect(Mser(n)/x+subs({x=y,y=x},Mser(n))/y-1/x*subs(x=0,Mser(n))-1/y*subs(x=0,y=x,Mser(n)),t,normal);

normal(series(%-%%,t,n+1));

IiIh

Now we want to write a functional equation for M, starting from the equation for A

eqM:=collect(subs(A(x,y)=op(2,AexprM), Am(1/x)=1/x*M(1/x,0), Am(1/y)=1/y*M(1/y,0),eqA),[M]);

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

Let us check this equation

indets(eqM);

PC1JInRHNiJJInhHRiRJInlHRiQtSSJNR0YkNiQiIiFGJS1GKDYkRipGJi1GKDYkRiVGJi1GKDYkRiZGJS1GKDYkKiRGJSEiIkYqLUYoNiRGM0YmLUYoNiQqJEYmRjRGKi1GKDYkRjlGJQ==

n:=6: normal(series(subs(M(0,x)=subs(x=0,y=x,Mser(n)), M(0,y)=subs(x=0,Mser(n)), M(x,y)=Mser(n), M(y,x)=subs({x=y,y=x},Mser(n)), M(1/x,0)=subs(x=1/x,y=0,Mser(n)), M(1/x,y)=subs(x=1/x,Mser(n)), M(1/y,0)=subs(y=0,x=1/y,Mser(n)), M(1/y,x)=subs({x=1/y,y=x},Mser(n)), eqM),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

The equation is symmetric in x and y

factor(eqM/subs({x=y,y=x},eqM));

IiIi

We take the negative part in x of eqM

eqM;

First, the negative part of terms not involving series in 1/x

normal(series(subs(M(1/x, 0)=0,M(1/x, y)=0,eqM),x,2));

KylJInhHNiIjIiIiIiIkISIjKihJInRHRiRGJiwoKiYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmNiNJIk1HRiQ2JCIiIUkieUdGJEYmRjlGJkYmLUY2NiRGOUY4RiYtRjY2JEY4RjghIiJGJkY5Rj5GPi1JIk9HRjJGNEY4

Then, the term in M(1/x,y) contains a constant term in x

series(subs(x=1/x,op(6,eqM)),x,3);

KytJInhHNiIqJkkidEdGJCIiIi1JIk1HRiQ2JCIiIUkieUdGJEYnRissJiomRiZGJy0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRic2I0YpRipGJ0YnKigsKComRiZGJylGLCIiI0YnRidGJkYnRiwhIiJGJ0YsRj1GKEYnRidGJywoKigjRidGPEYnRiZGJy0tJkYyNiRGJ0YnRjdGKkYnRidGJUYnKihGOUYnRixGPUYvRidGJ0Y8LUkiT0dGNEY2IiIk

eqMxneg:=convert(%%,polynom)+op(5,eqM)+op(6,eqM)-t*M(0,y);

LCwqJiMiIiIiIiRGJSlJInhHNiIiIiMhIiJGJSoqSSJ0R0YpRiUsKComLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGJTYjSSJNR0YpNiQiIiFJInlHRilGJUY8RiVGJS1GOTYkRjxGO0YlLUY5NiRGO0Y7RitGJUY8RitGKEYrRiUqKkYtRiVGPEYrRihGKy1GOTYkKiRGKEYrRjtGJUYrKiosLCooRi1GJUYnRiVGPEYlRiUqKEYtRiVGKEYlKUY8RipGJUYlKiZGLUYlRihGJUYlKiZGLUYlRjxGJUYlKiZGKEYlRjxGJUYrRiVGJ0YrRjxGKy1GOTYkRkRGPEYlRiUqJkYtRiUtRjlGOkYlRis=

indets(eqMxneg);

PCpJInRHNiJJInhHRiRJInlHRiQtSSJNR0YkNiQiIiFGJi1GKDYkRiZGKi1GKDYkKiRGJSEiIkYqLUYoNiRGL0YmLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiI2I0YoRik=

paper version

eqpap:=-t/x*((D[1](M))(0, y)+M(y,0)/y-M(0,0)/y)+K*M(1/x,y)/x+t*M(0,y)-1/3/x^2+t/x/y*M(1/x,0);

LCwqKEkidEc2IiIiIkkieEdGJSEiIiwoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJjYjSSJNR0YlNiQiIiFJInlHRiVGJiomLUYzNiRGNkY1RiZGNkYoRiYqJi1GMzYkRjVGNUYmRjZGKEYoRiZGKCooLCZGJkYmKiZGJEYmLCpGJ0YmKiRGJ0YoRiZGNkYmKiRGNkYoRiZGJkYoRiYtRjM2JEZBRjZGJkYnRihGJiomRiRGJi1GM0Y0RiZGJiomI0YmIiIkRiYpRiciIiNGKEYoKipGJEYmRjZGKEYnRigtRjM2JEZBRjVGJkYm

collect(eqpap+eqMxneg,[(D[1](M))(0, y)],factor);

IiIh

Let us check that what remains is non-negative in x...

normal(series(normal(eqM-eqMxneg),x,3));

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

... and that the negative part is indeed negative in x

normal(series(subs(x=1/x,eqMxneg),x,3));

KylJInhHNiIqJiwuKigtSSJNR0YkNiQiIiFJInlHRiQiIiJJInRHRiRGLSlGLCIiI0YtRi0qKkYwRi1GLkYtLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLTYjRilGKkYtRixGLUYtKiYtRik2JEYsRitGLUYuRi1GLSooRjBGLS1GKTYkRitGK0YtRi5GLSEiIiomRi5GLUYoRi1GLSomRixGLUYoRi1GQUYtRixGQUYtLCQqKCNGLSIiJ0YtLDAqKkZHRi1GLkYtRjJGLUYvRi1GQSoqIiIkRi1GLkYtLS0mRjU2JEYtRi1GOkYqRi1GLEYtRkEqKkZHRi1GLkYtRihGLUYsRi1GQSooRkdGLUYuRi0tRjNGQEYtRi0qKEZHRi1GLkYtRjJGLUZBKihGR0YtRjJGLUYsRi1GLSomRjBGLUYsRi1GQUYtRixGQUZBRjAtSSJPR0Y3RjlGSw==

The coefficient of 1/x in eqMxneg

com1:=coeff(%,x):

paper version

eqpap:=-t*((D[1](M))(0, y)+M(y,0)/y-M(0,0)/y)+(1-t*(y+1/y))*M(0,y)-t*(D[1](M))(0, y)+t*M(0,0)/y;

LCoqJkkidEc2IiIiIiwoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJjYjSSJNR0YlNiQiIiFJInlHRiVGJiomLUYxNiRGNEYzRiZGNCEiIkYmKiYtRjE2JEYzRjNGJkY0RjhGOEYmRjgqJiwmRiZGJiomRiRGJiwmRjRGJiokRjRGOEYmRiZGOEYmLUYxRjJGJkYmKiZGJEYmRihGJkY4KihGJEYmRjRGOEY6RiZGJg==

factor(com1/eqpap);

ISIi

solve(com1,(D[1](M))(0, y));

LCQqKiMiIiIiIiNGJSwsKigtSSJNRzYiNiQiIiFJInlHRitGJUkidEdGK0YlKUYuRiZGJUYlKiYtRio2JEYuRi1GJUYvRiVGJSooRiZGJS1GKjYkRi1GLUYlRi9GJSEiIiomRi9GJUYpRiVGJSomRi5GJUYpRiVGN0YlRi5GN0YvRjdGNw==

eqMxneg:=collect(subs((D[1](M))(0, y)=%,eqMxneg),M,factor);

LCwqLCMiIiIiIiNGJSwqKipGJkYlSSJ0RzYiRiVJInhHRipGJUkieUdGKkYlRiUqJkYpRiUpRixGJkYlRiVGKUYlRiwhIiJGJUYrRi9GLEYvLUkiTUdGKjYkIiIhRixGJUYvKiYjRiUiIiRGJSlGK0YmRi9GJSosRiRGJUYpRiVGLEYvLUYxNiRGLEYzRiVGK0YvRiUqKkYpRiVGLEYvRitGLy1GMTYkKiRGK0YvRjNGJUYvKiosLCooRilGJUY3RiVGLEYlRiUqKEYpRiVGK0YlRi5GJUYlKiZGKUYlRitGJUYlKiZGKUYlRixGJUYlKiZGK0YlRixGJUYvRiVGN0YvRixGLy1GMTYkRj5GLEYlRiU=

eqpap:=K*(2*M(1/x,y)-M(0,y))-2/3/x+2*t/y*M(1/x,0)-t*(1/x-x)*M(0,y)-t*M(y,0)/y;

LCwqJiwmIiIiRiUqJkkidEc2IkYlLCpJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiVGJUYsRiUsJiomIiIjRiUtSSJNR0YoNiRGK0YtRiVGJS1GMzYkIiIhRi1GLEYlRiUqJiNGMSIiJEYlRipGLEYsKipGMUYlRidGJUYtRiwtRjM2JEYrRjdGJUYlKihGJ0YlLCZGK0YlRipGLEYlRjVGJUYsKihGJ0YlRi1GLC1GMzYkRi1GN0YlRiw=

factor(eqMxneg/eqpap);

LCQqJiMiIiIiIiNGJUkieEc2IiEiIkYp

eqcatM:=map(factor,subs(x=1/x,eqpap));

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRiZGKEYqRihGKComRiZGKEYsRihGKComRipGKEYsRighIiJGKCwmKiZGK0YoLUkiTUdGJzYkRipGLEYoRigtRjY2JCIiIUYsRjJGKEYqRjJGLEYyRjIqJiNGKyIiJEYoRipGKEYyKipGK0YoRiZGKEYsRjItRjY2JEYqRjpGKEYoKixGJkYoLCZGKkYoRihGMkYoLCZGKkYoRihGKEYoRjhGKEYqRjJGMiooRiZGKEYsRjItRjY2JEYsRjpGKEYy

Let us check

indets(eqcatM);

PClJInRHNiJJInhHRiRJInlHRiQtSSJNR0YkNiQiIiFGJi1GKDYkRiVGKi1GKDYkRiVGJi1GKDYkRiZGKg==

n:=7: normal(series(subs( M(0,0)=subs(x=0,y=0,Mser(n)), M(x,0)=subs(y=0,Mser(n)),M(0,y)=subs(x=0,Mser(n)),M(x,y)=Mser(n), M(y,0)=subs(y=0,x=y,Mser(n)),M0=subs(x=0,y=0,Mser(n)), eqcatM),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Checking if this implies the functional equation eqM coming from that of A(x,y)

Mxysol:=solve(eqcatM,M(x,y));

LCQqKCMiIiIiIidGJSwwKiwiIiRGJS1JIk1HNiI2JCIiIUkieUdGLEYlSSJ0R0YsRiVJInhHRixGJSlGLyIiI0YlRiUqKkYpRiVGKkYlRjBGJUYxRiVGJSoqRiZGJUYwRiVGKkYlRi9GJUYlKipGKUYlRipGJUYxRiVGL0YlISIiKipGJkYlRjBGJS1GKzYkRjFGLkYlRjFGJUYlKipGKUYlRjBGJS1GKzYkRi9GLkYlRjFGJUY3KihGM0YlKUYxRjNGJUYvRiVGN0YlLCwqKEYwRiVGP0YlRi9GJUYlKihGMEYlRjFGJUYyRiVGJSomRjBGJUYxRiVGJSomRjBGJUYvRiVGJSomRjFGJUYvRiVGN0Y3RiU=

factor(subs(M(x,y)=Mxysol,M(y,x)=subs({x=y,y=x},Mxysol),M(1/y,x)=subs({x=1/y,y=x},Mxysol),M(1/x,y)=subs(x=1/x,Mxysol), eqM));

IiIh

Results of the above sub-section

PexprM:=M(x, y)/x-M(0, y)/x+M(y, x)/y-M(0, x)/y;

LCoqJi1JIk1HNiI2JEkieEdGJkkieUdGJiIiIkYoISIiRioqJi1GJTYkIiIhRilGKkYoRitGKyomLUYlNiRGKUYoRipGKUYrRioqJi1GJTYkRi9GKEYqRilGK0Yr

eqcatM:=-(t*x^2*y+t*x*y^2+t*x+t*y-x*y)*(2*M(x, y)-M(0, y))/(x*y)-(2/3)*x+2*t*M(x, 0)/y-t*(x-1)*(x+1)*M(0, y)/x-t*M(y, 0)/y;

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRiZGKEYqRihGKComRiZGKEYsRihGKComRipGKEYsRighIiJGKCwmKiZGK0YoLUkiTUdGJzYkRipGLEYoRigtRjY2JCIiIUYsRjJGKEYqRjJGLEYyRjIqJiNGKyIiJEYoRipGKEYyKipGK0YoRiZGKC1GNjYkRipGOkYoRixGMkYoKixGJkYoLCZGKkYoRihGMkYoLCZGKkYoRihGKEYoRjhGKEYqRjJGMiooRiZGKC1GNjYkRixGOkYoRixGMkYy

Write N(x,y)=2*M(x, y)-M(0, y)

eqcatN :=-(t*x^2*y+t*x*y^2+t*x+t*y-x*y)*N(x, y)/(x*y)-(2/3)*x+2*t*M(x, 0)/y-t*(x-1)*(x+1)*M(0, y)/x-t*M(y, 0)/y;

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRiZGKEYqRihGKComRiZGKEYsRihGKComRipGKEYsRighIiJGKC1JIk5HRic2JEYqRixGKEYqRjJGLEYyRjIqJiNGKyIiJEYoRipGKEYyKipGK0YoRiZGKC1JIk1HRic2JEYqIiIhRihGLEYyRigqLEYmRigsJkYqRihGKEYyRigsJkYqRihGKEYoRigtRjs2JEY9RixGKEYqRjJGMiooRiZGKC1GOzYkRixGPUYoRixGMkYy

The roots of the kernel

Yens:=factor([solve(K,y)]);

NyQsJCoqIyIiIiIiI0YmSSJ0RzYiISIiSSJ4R0YpRiosKiomRihGJilGK0YnRiZGKkYoRipGK0YmKiQpKiYsKkYtRiYqKEYnRiZGKEYmRitGJkYmRihGJkYrRipGJiwqRi1GJkYzRipGKEYmRitGKkYmRiVGJkYmRiZGJiwkKipGJUYmLCpGLUYmRi9GJkYoRiZGK0YqRiZGKEYqRitGKkYq

The series Y0 is denoted by Y in the paper

Y0sol:=op(2,Yens);Y1sol:=op(1,Yens);

LCQqKiMiIiIiIiNGJSwqKiZJInRHNiJGJSlJInhHRipGJkYlRiUqJCkqJiwqRihGJSooRiZGJUYpRiVGLEYlRiVGKUYlRiwhIiJGJSwqRihGJUYxRjJGKUYlRixGMkYlRiRGJUYlRilGJUYsRjJGJUYpRjJGLEYyRjI=

LCQqKiMiIiIiIiNGJUkidEc2IiEiIkkieEdGKEYpLCoqJkYnRiUpRipGJkYlRilGJ0YpRipGJSokKSomLCpGLEYlKihGJkYlRidGJUYqRiVGJUYnRiVGKkYpRiUsKkYsRiVGMkYpRidGJUYqRilGJUYkRiVGJUYlRiU=

Delta:=(1-t*(x+1)*(1+1/x))*(1+t*(1-x)*(1-1/x));

KiYsJiIiIkYkKihJInRHNiJGJCwmSSJ4R0YnRiRGJEYkRiQsJkYkRiQqJEYpISIiRiRGJEYsRiQsJkYkRiQqKEYmRiQsJkYkRiRGKUYsRiQsJkYkRiRGK0YsRiRGJEYk

factor(Delta+4*t^2);

KiYpLCgqJkkidEc2IiIiIilJInhHRiciIiNGKEYoRiZGKEYqISIiRitGKEYpRiw=

version of the paper

Ypap:=(1-t*(x+1/x)-sqrt(Delta))/2/t;

LCQqKCMiIiIiIiNGJSwoRiVGJSomSSJ0RzYiRiUsJkkieEdGKkYlKiRGLCEiIkYlRiVGLiokKSomLCZGJUYlKihGKUYlLCZGLEYlRiVGJUYlLCZGJUYlRi1GJUYlRi5GJSwmRiVGJSooRilGJSwmRiVGJUYsRi5GJSwmRiVGJUYtRi5GJUYlRiVGJEYlRi5GJUYpRi5GJQ==

simplify(Ypap/Y0sol,symbolic);

IiIi

Y0ser:=proc(n) simplify(series(Y0sol,t,n+2),symbolic): end:

Y0ser(4);

Ky1JInRHNiIiIiJGJSomLCYqJClJInhHRiQiIiNGJUYlRiVGJUYlRiohIiJGKyomRilGLCwoKiQpRioiIiVGJUYlKiYiIiRGJUYpRiVGJUYlRiVGJUYzKigpRipGM0YsRidGJSwoRi9GJSomIiImRiVGKUYlRiVGJUYlRiVGMS1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY4

factor(expand(Y0sol*Y1sol));

IiIi

<Text-field style="Heading 2" layout="Heading 2">Canceling the kernel: an equation between M(x,0), M(0,x) and M(0,1/x)</Text-field>

Pairs than cancel the kernel : (x,Y), (1/x,Y), (Y,x) et (Y, 1/x)

f1:=collect(subs(N(x,y)=0,y=Y,-eqcatN),[M]); f2:=subs(x=1/x,f1);

LCoqJiMiIiMiIiQiIiJJInhHNiJGJ0YnKipGJUYnSSJ0R0YpRictSSJNR0YpNiRGKCIiIUYnSSJZR0YpISIiRjEqLEYrRicsJkYoRidGJ0YxRicsJkYoRidGJ0YnRictRi02JEYvRjBGJ0YoRjFGJyooRitGJy1GLTYkRjBGL0YnRjBGMUYn

LCoqJiMiIiMiIiQiIiJJInhHNiIhIiJGJyoqRiVGJ0kidEdGKUYnLUkiTUdGKTYkKiRGKEYqIiIhRidJIllHRilGKkYqKixGLEYnLCZGMEYnRidGKkYnLCZGJ0YnRjBGJ0YnLUYuNiRGMUYyRidGKEYnRicqKEYsRictRi42JEYyRjFGJ0YyRipGJw==

f3:=collect(subs({x=y,y=x},N(y,x)=0,y=Y,-eqcatN),[M]); f4:=subs(x=1/x,f3);

LCoqJiMiIiMiIiQiIiJJIllHNiJGJ0YnKipGJUYnSSJ0R0YpRictSSJNR0YpNiRGKCIiIUYnSSJ4R0YpISIiRjEqLEYrRicsJkYoRidGJ0YxRicsJkYoRidGJ0YnRictRi02JEYvRjBGJ0YoRjFGJyooRitGJy1GLTYkRjBGL0YnRjBGMUYn

LCoqJiMiIiMiIiQiIiJJIllHNiJGJ0YnKipGJUYnSSJ0R0YpRictSSJNR0YpNiRGKCIiIUYnSSJ4R0YpRichIiIqLEYrRicsJkYoRidGJ0YxRicsJkYoRidGJ0YnRictRi02JEYvKiRGMEYxRidGKEYxRicqKEYrRictRi02JEY3Ri9GJ0YwRidGJw==

indets(f1*f2*f3*f4);

PCtJIllHNiJJInRHRiRJInhHRiQtSSJNR0YkNiQiIiFGIy1GKDYkRipGJi1GKDYkRioqJEYmISIiLUYoNiRGI0YqLUYoNiRGJkYqLUYoNiRGL0Yq

We eliminate the series involving Y

MYsol:=factor(solve({f1,f2},{M(0,Y),M(Y,0)}));

PCQvLUkiTUc2IjYkIiIhSSJZR0YmLCQqLiMiIiIiIiRGLSwqKipGLkYtSSJ0R0YmRi0tRiU2JCokSSJ4R0YmISIiRihGLUY1Ri1GLSoqRi5GLUYxRi0tRiU2JEY1RihGLUY1Ri1GNiomRilGLSlGNSIiI0YtRi1GKUY2Ri1GKUY2RjFGNiwmRjVGLUYtRjZGNiwmRjVGLUYtRi1GNkY2Ly1GJTYkRilGKCwkKipGLEYtLCpGMEYtRjdGLUY6RjZGKUY2Ri1GNUY2RjFGNkYt

We thus obtain two equations between four series in x

f3bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f3))),[M],factor);

LCoqLEkieEc2IiIiIiwmSSJZR0YlRiZGJiEiIkYmLCZGKEYmRiZGJkYmRihGKS1JIk1HRiU2JCIiIUYkRiZGJi1GLDYkRiRGLkYpKiYiIiNGJi1GLDYkKiRGJEYpRi5GJkYpKiwjRjIiIiRGJkYoRiYsJiomRjJGJilGJEYyRiZGJkYmRiZGJkkidEdGJUYpRiRGKUYm

f4bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f4))),[M],factor);

LCoqLCwmSSJZRzYiIiIiRichIiJGJywmRiVGJ0YnRidGJ0YlRihJInhHRiZGKC1JIk1HRiY2JCIiISokRipGKEYnRicqJiIiI0YnLUYsNiRGKkYuRidGKC1GLDYkRi9GLkYoKiwjRjEiIiRGJ0YlRicsJiokKUYqRjFGJ0YnRjFGJ0YnSSJ0R0YmRihGKkYoRic=

These two equations just differ by x-> 1/x

factor(%%/subs(x=1/x,%));

IiIi

Checking:

n:=10: normal(series(subs(M(0,x)=subs(x=0,y=x,Mser(n)),M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(x,0)=subs(y=0,Mser(n)), M(1/x,0)=subs(x=1/x,y=0,Mser(n)), M10=coeff(subs(y=0,Mser(n)),x,1),Y=Y0ser(n+2),f3bis),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

Eliminate M(1/x,0)

f8:=collect(f3bis-2*f4bis,[M],factor);

LCoqLEkieEc2IiIiIiwmSSJZR0YlRiZGJiEiIkYmLCZGKEYmRiZGJkYmRihGKS1JIk1HRiU2JCIiIUYkRiZGJiouIiIjRiZGJ0YmRipGJkYoRilGJEYpLUYsNiRGLiokRiRGKUYmRikqJiIiJEYmLUYsNiRGJEYuRiZGJioqRjBGJkYoRiZGJEYpSSJ0R0YlRilGKQ==

What if we eliminate another of the four series ? For instance M(0,1/x) ? This is just f3bis

What we should remember from the above subsection

f8:= x*(Y-1)*(Y+1)*M(0, x)/Y-(2*(Y-1))*(Y+1)*M(0, 1/x)/(Y*x)+3*M(x, 0)-2*Y/(x*t);

LCoqLEkieEc2IiIiIiwmSSJZR0YlRiZGJiEiIkYmLCZGKEYmRiZGJkYmLUkiTUdGJTYkIiIhRiRGJkYoRilGJiouIiIjRiZGJ0YmRipGJi1GLDYkRi4qJEYkRilGJkYoRilGJEYpRikqJiIiJEYmLUYsNiRGJEYuRiZGJioqRjBGJkYoRiZGJEYpSSJ0R0YlRilGKQ==

factor(expand(rationalize(subs(Y=Y0sol,Y-1/Y))));

LCQqKEkidEc2IiEiIkkieEdGJUYmKSomLCoqJkYkIiIiKUYnIiIjRixGLCooRi5GLEYkRixGJ0YsRixGJEYsRidGJkYsLCpGK0YsRi9GJkYkRixGJ0YmRiwjRixGLkYsRiY=

simplify(%/sqrt(Delta),symbolic);

LCQqJEkidEc2IiEiIkYm

Introducing De=Delta

f8form:=1/t*sqrt(De)*(x*M(0,x)-2/x*M(0,1/x))-3*M(x, 0)+2*Y/(x*t);

LCgqKEkidEc2IiEiIilJI0RlR0YlIyIiIiIiI0YqLCYqJkkieEdGJUYqLUkiTUdGJTYkIiIhRi5GKkYqKihGK0YqRi5GJi1GMDYkRjIqJEYuRiZGKkYmRipGKiomIiIkRiotRjA2JEYuRjJGKkYmKipGK0YqSSJZR0YlRipGLkYmRiRGJkYq

Checking

n:=10: normal(series(subs(De=Delta, M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(0,x)=subs(x=0,y=x,Mser(n)),M(x,0)=subs(y=0,Mser(n)), Y=Y0ser(n+2),f8form),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

Rser:=proc (n) collect(t*subs(y = 0, Mser(n-1)), t, normal) end proc:

Sser:=proc(n) collect(subs(x=0,y=x,Mser(n)*y*t),t) end proc:

f13 := (S(x)-2*S(1/x)-1/x)*sqrt(De)-3*t*R(x)+1/x-t-t/x^2;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGL0YwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjBGL0YqRjZGMComRjZGKilGKUYsRjBGMA==

<Text-field style="Heading 2" layout="Heading 2">An equation between M(0,x) and M(0,1/x) (or S(x) and S(1/x))</Text-field>

The following series is non-negative in x

f12:=sqrt(Delta)*(x*M(0,x)-2/x*M(0,1/x))+(1-t*(x+1/x)-sqrt(Delta))/x/t;

LCYqJikqJiwmIiIiRicqKEkidEc2IkYnLCZJInhHRipGJ0YnRidGJywmRidGJyokRiwhIiJGJ0YnRi9GJywmRidGJyooRilGJywmRidGJ0YsRi9GJywmRidGJ0YuRi9GJ0YnRicjRiciIiNGJywmKiZGLEYnLUkiTUdGKjYkIiIhRixGJ0YnKihGNUYnRixGLy1GOTYkRjtGLkYnRi9GJ0YnKigsKEYnRicqJkYpRicsJkYsRidGLkYnRidGLyokRiRGJ0YvRidGLEYvRilGL0Yn

and this one is even positive in x

f12plus:=sqrt(Delta)*(x*M(0,x)-2/x*M(0,1/x))+(1-t*(x+1/x)-sqrt(Delta))/x/t-3*t*M00;

LCgqJikqJiwmIiIiRicqKEkidEc2IkYnLCZJInhHRipGJ0YnRidGJywmRidGJyokRiwhIiJGJ0YnRi9GJywmRidGJyooRilGJywmRidGJ0YsRi9GJywmRidGJ0YuRi9GJ0YnRicjRiciIiNGJywmKiZGLEYnLUkiTUdGKjYkIiIhRixGJ0YnKihGNUYnRixGLy1GOTYkRjtGLkYnRi9GJ0YnKigsKEYnRicqJkYpRicsJkYsRidGLkYnRidGLyokRiRGJ0YvRidGLEYvRilGL0YnKigiIiRGJ0YpRidJJE0wMEdGKkYnRi8=

Checking these two properties

n:=10: normal(series(subs( M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(0,x)=subs(x=0,y=x,Mser(n)),f12),t,n));

KzVJInRHNiJJInhHRiQiIiIsJiokKUYlIiIjRiZGJiIiJEYmRioqJkYlRiYsJkYoRiYiIiZGJkYmRissKCokKUYlIiIlRiZGJiomIiIpRiZGKUYmRiYiI0BGJkYyKiZGJUYmLChGMEYmKiYiIzdGJkYpRiZGJiIjVkYmRiZGLiwqKiQpRiUiIidGJkYmKiYiIzxGJkYxRiZGJiomIiMkKUYmRilGJkYmIiQmPkYmRj4qJkYlRiYsKkY8RiYqJiIjQkYmRjFGJkYmKiYiJF0iRiZGKUYmRiYiJFUlRiZGJiIiKCwsKiQpRiVGNEYmRiYqJiIjSUYmRj1GJkYmKiYiJGIjRiZGMUYmRiYqJiIkYSpGJkYpRiZGJiIldz9GJkY0LUkiT0clKnByb3RlY3RlZEc2I0YmIiIq

n:=10: normal(series(subs( M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(0,x)=subs(x=0,y=x,Mser(n)),M00=subs(x=0,y=0,Mser(n)),f12plus),t,n));

KzVJInRHNiJJInhHRiQiIiIqJClGJSIiI0YmRikqJkYlRiYsJkYnRiYiIiZGJkYmIiIkKiZGKEYmLCZGJ0YmIiIpRiZGJiIiJSomRiVGJiwoKiQpRiVGMUYmRiYqJiIjN0YmRihGJkYmIiNWRiZGJkYsKiZGKEYmLChGNEYmKiYiIzxGJkYoRiZGJiIjJClGJkYmIiInKiZGJUYmLCoqJClGJUY+RiZGJiomIiNCRiZGNUYmRiYqJiIkXSJGJkYoRiZGJiIkVSVGJkYmIiIoKiZGKEYmLCpGQUYmKiYiI0lGJkY1RiZGJiomIiRiI0YmRihGJkYmIiRhKkYmRiZGMC1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIiKg==

subs(x=0,%);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

Introduce the series R and S

Rser:=proc(n) collect(t*subs(y=0,Mser(n-1)),t,normal): end:

Sser:=proc(n) collect(subs(x=0,y=x,Mser(n)*y*t),t); end:

Rser(4);Sser(4);

LCoqJiwoIiIoIiIiKiYjIiIpIiIkRiYpSSJ4RzYiIiIjRiZGJiomI0YmRipGJilGLCIiJUYmRiZGJilJInRHRi1GMkYmRiYqJiwmKiYjIiImRipGJkYsRiZGJiomRjBGJilGLEYqRiZGJkYmKUY0RipGJkYmKiYsJkYmRiYqJkYwRiZGK0YmRiZGJilGNEYuRiZGJiooRjBGJkYsRiZGNEYmRiY=

LCoqKEkieEc2IiIiIiwmKiYiIiVGJilGJCIiJEYmRiYqJiIjQUYmRiRGJkYmRiYpSSJ0R0YlIiImRiZGJiooRiRGJiwmKiZGK0YmKUYkIiIjRiZGJiIiKEYmRiYpRi9GKUYmRiYqKEY1RiYpRi9GK0YmRjRGJkYmKiZGJEYmKUYvRjVGJkYm

f13:=collect(subs(M(0,x)=S(x)/t/x,M(0,1/x)=x/t*S(1/x), M(x,0)=R(x)/t,Y=(1-t*(x+1/x)-sqrt(De))/2/t,t^2*f8form),[De,R],expand);

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGL0YwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjBGL0YqRjZGMComKUYpRixGMEY2RipGMA==

Checking once more

n:=10: normal(series(subs(De=Delta, S(x)=Sser(n), S(1/x)=subs(x=1/x,Sser(n)),R(x)=Rser(n),f13),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

op(1,f13);

KiYsKC1JIlNHNiI2I0kieEdGJiIiIiomIiIjRiktRiU2IyokRighIiJGKUYvRi5GL0YpKUkjRGVHRiYjRilGK0Yp

u0:=op(1,f13)^2-subs(De=0,f13)^2;

LCYqJiksKC1JIlNHNiI2I0kieEdGKCIiIiomIiIjRistRic2IyokRiohIiJGK0YxRjBGMUYtRitJI0RlR0YoRitGKyokKSwqKigiIiRGK0kidEdGKEYrLUkiUkdGKEYpRitGMUYwRitGOEYxKiYpRipGLUYxRjhGK0YxRi1GK0Yx

u1:=collect(u0,[S,R],distributed,expand);

LD4qJkkjRGVHNiIiIiIpLUkiU0dGJTYjSSJ4R0YlIiIjRiZGJioqIiIlRiZGJEYmRihGJi1GKTYjKiRGKyEiIkYmRjIqKkYsRiZGK0YyRiRGJkYoRiZGMiooRi5GJkYkRiYpRi9GLEYmRiYqKkYuRiZGK0YyRiRGJkYvRiZGJiooIiIqRiYpSSJ0R0YlRixGJiktSSJSR0YlRipGLEYmRjIqJiwoKigiIidGJkY6RiZGK0YyRiYqJkZBRiZGOUYmRjIqKEZBRiYpRitGLEYyRjlGJkYyRiZGPEYmRiYqJkZERjJGJEYmRiYqJEZERjJGMiooRixGJkY6RiZGK0YyRiYqKEYsRiYpRisiIiRGMkY6RiZGJiokRjlGJkYyKihGLEYmRkRGMkY5RiZGMiomKUYrRi5GMkY5RiZGMg==

u1expl:=collect(subs(De=Delta,u0),[S,R],distributed,expand);

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

Extract the negative part in x (which must be zero)

expand(Delta);

LC4iIiJGIyooIiIjRiNJInRHNiJGI0kieEdGJyEiIkYpKihGJUYjRihGI0YmRiNGKSomRiVGIylGJkYlRiNGKSomRixGIylGKEYlRiNGIyomRi5GKUYsRiNGIw==

First, the contribution of the series in x

normal(subs(S(0)=0,series(subs(S(1/x)=0,u1expl),x,3)));

KylJInhHNiIsKCooIiIjIiIiKUkidEdGJEYnRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJTR0YkNiMiIiFGKCEiIiooIiInRihGKUYoLUkiUkdGJEYzRihGNSomIiIlRihGKUYoRjUhIiMsKiomRilGKC0tLUkjQEBHRi42JEYtRidGMUYzRihGNSooRjtGKEYqRihGK0YoRigqKEY3RihGKUYoLS1GLTYjRjlGM0YoRjUqKEY3RihGKkYoRjhGKEYoRjUtSSJPR0YvNiNGKEY0

Then, the series in 1/x still give rise to a constant term in x

subs(S(x)=0,R(x)=0,u1expl)+4*t^2/x^2;

LCYqJiwuIiIlIiIiKigiIilGJkkidEc2IkYmSSJ4R0YqISIiRiwqKEYoRiZGK0YmRilGJkYsKiZGKEYmKUYpIiIjRiZGLCooRiVGJkYvRiYpRitGMEYmRiYqKEYlRiZGMkYsRi9GJkYmRiYpLUkiU0dGKjYjKiRGK0YsRjBGJkYmKiYsLiomRiVGJkYrRixGJiooRihGJkYyRixGKUYmRiwqJkYoRiZGKUYmRiwqKEYoRiZGK0YsRi9GJkYsKihGJUYmRitGJkYvRiZGJiooRiVGJilGKyIiJEYsRi9GJkYmRiZGNUYmRiY=

normal(subs(S(0)=0,series(subs(x=1/x,%),x,3)));

KydJInhHNiIsJiooIiIlIiIiKUkidEdGJCIiI0YoKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJIlNHRiQ2IyIiIUYrRihGKCooRidGKEYpRihGLUYoRihGNi1JIk9HRjE2I0YoRig=

The negative part of u1

u2:=-4*Sneg(1/x)+4*De*(S(1/x)^2+1/x*S(1/x)) -convert(%,polynom) +convert(%%%,polynom);

LC4qJiIiJSIiIi1JJVNuZWdHNiI2IyokSSJ4R0YoISIiRiVGLCooRiRGJUkjRGVHRihGJSwmKiQpLUkiU0dGKEYpIiIjRiVGJSomRitGLEYyRiVGJUYlRiUqKEYkRiUpSSJ0R0YoRjRGJSktLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRjM2IyIiIUY0RiVGLCooRiRGJUY3RiVGOkYlRiwqJiwoKihGNEYlRjdGJUY6RiVGLCooIiInRiVGN0YlLUkiUkdGKEZBRiVGLComRiRGJUY3RiVGLEYlKUYrRjRGLEYlKiYsKiomRjdGJS0tLUkjQEBHRj02JEY8RjRGQEZBRiVGLCooRiRGJUY4RiVGOkYlRiUqKEZIRiVGN0YlLS1GPDYjRkpGQUYlRiwqKEZIRiVGOEYlRklGJUYlRiVGK0YsRiU=

collect(u2,[De,Sneg,x],factor);

LCwqJiwmKiYiIiUiIiIpLUkiU0c2IjYjKiRJInhHRishIiIiIiNGJ0YnKihGJkYnRi5GL0YpRidGJ0YnSSNEZUdGK0YnRicqJkYmRictSSVTbmVnR0YrRixGJ0YvKipGJkYnKUkidEdGK0YwRictLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjRio2IyIiIUYnLCZGOUYnRidGJ0YnRi8qKEY4RicsKiomLS0tSSNAQEdGPDYkRjtGMEY/RkBGJ0Y4RidGJyooIiInRidGOEYnLS1GOzYjSSJSR0YrRkBGJ0YnKiZGJkYnRjlGJ0YvKiZGTEYnLUZQRkBGJ0YvRidGLkYvRi8qKkYwRidGN0YnLChGOUYnKiYiIiRGJ0ZTRidGJ0YwRidGJylGLkYwRi9GLw==

u3:=map(factor,subs(D(S)(0)=R(0),%/4));

LCwqKi1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCYqJkYkRitGKUYrRitGK0YrRitJI0RlR0YmRitGKUYqRistSSVTbmVnR0YmRidGKiooKUkidEdGJiIiI0YrLUkiUkdGJjYjIiIhRissJkY1RitGK0YrRitGKioqI0YrIiIlRitGM0YrLCgqJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2JEkiREdGQ0Y0NiNGJUY3RitGM0YrRisqKCIiJ0YrRjNGKy0tRkc2I0Y2RjdGK0YrKiYiIzVGK0Y1RitGKkYrRilGKkYqKihGMkYrLCYqJkY0RitGNUYrRitGK0YrRispRilGNEYqRio=

let us check: this must be non-negative

n:=10: expand(series(subs(De=Delta,Sneg(1/x)=Delta*S(x)*S(1/x),S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,R(0)=subs(x=0,Rser(n)), (D(R))(0)=coeff(Rser(n),x,1),(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u3),t,n));

Ky9JInRHNiIhIiIiIiUhIikiIicsJComIiIjIiIiSSJ4R0YkRixGLCIiKCwmKiYiIiRGLClGLUYrRixGLCIjeUYlIiIpLCYqJkYmRiwpRi1GMUYsRiwqJiIjT0YsRi1GLEYsIiIqLUkiT0clKnByb3RlY3RlZEc2I0YsIiM1

and this must be negative

n:=14: expand(series(subs(De=Delta,Sneg(1/x)=0,S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,R(0)=subs(x=0,Rser(n)), (D(R))(0)=coeff(Rser(n),x,1),(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u3),t,n));

KzNJInRHNiIsJComIiIjIiIiSSJ4R0YkISIiRioiIigsJComIiIkRigpRilGJ0YqRioiIiksJiomIiNPRihGKUYqRioqJiIiJUYoKUYpRi5GKkYqIiIqLCYqJiIjbEYoRi9GKkYqKiYiIiZGKClGKUY1RipGKiIjNSwoKiYiJEkmRihGKUYqRioqJiIkLyJGKEY2RipGKiomIiInRigpRilGPEYqRioiIzYsKComIiV6NUYoRi9GKkYqKiYiJGIiRihGPUYqRioqJkYrRigpRilGRUYqRioiIzcsKiomIiVDdUYoRilGKkYqKiYiJWM+RihGNkYqRioqJiIkPyNGKEZGRipGKiomRjBGKClGKUYrRipGKiIjOC1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjOQ==

u3form:=S(1/x)*(S(1/x)*x+1)*De/x-Sneg(1/x)-F0-F1/x-F2/x^2;

LCwqKi1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCYqJkYkRitGKUYrRitGK0YrRitJI0RlR0YmRitGKUYqRistSSVTbmVnR0YmRidGKkkjRjBHRiZGKiomSSNGMUdGJkYrRilGKkYqKiZJI0YyR0YmRispRikiIiNGKkYq

F0expr:=-coeff(subs(Sneg(1/x)=0,De=0,u3),x,0); F1expr:=-coeff(subs(Sneg(1/x)=0,De=0,u3),x,-1); F2expr:=-coeff(subs(Sneg(1/x)=0,De=0,u3),x,-2);

KigpSSJ0RzYiIiIjIiIiLUkiUkdGJTYjIiIhRicsJkYoRidGJ0YnRic=

LCQqKCMiIiIiIiVGJUkidEc2IkYlLCgqJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkiREdGLyIiIzYjSSJTR0YoNiMiIiFGJUYnRiVGJSooIiInRiVGJ0YlLS1GMzYjSSJSR0YoRjdGJUYlKiYiIzVGJS1GPkY3RiUhIiJGJUYl

KiYpSSJ0RzYiIiIjIiIiLCYqJkYmRictSSJSR0YlNiMiIiFGJ0YnRidGJ0Yn

We now reconstruct Delta S(x) S(1/x)

u5:=collect((factor( De*S(x)*S(1/x)-solve(u3,Sneg(1/x))-P0-subs(x=1/x,solve(u3,Sneg(1/x)) ))),[De],expand);

LD4qJiwsKiQpLUkiU0c2IjYjKiRJInhHRikhIiIiIiMiIiJGLSomRidGLy1GKDYjRixGL0YvKiQpRjFGLkYvRi0qJkYsRi9GMUYvRi0qJkYsRi1GJ0YvRi1GL0kjRGVHRilGL0YvKipGLkYvKUkidEdGKUYuRi8tSSJSR0YpNiMiIiFGLylGLEYuRi9GLyoqI0YvIiIlRi8tLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiRJIkRHRkdGLjYjRihGPUYvRjlGL0YsRi9GLyooRi5GL0Y5Ri8pRjtGLkYvRi8qKiMiIiRGLkYvLS1GSzYjRjxGPUYvRjlGL0YsRi9GLyomRjlGL0Y/Ri9GLyooRi5GL0Y5Ri9GO0YvRi8qKiMiIiZGLkYvRjtGL0Y6Ri9GLEYvRi0qKkZBRi9GQ0YvRjlGL0YsRi1GLyoqRlBGL0ZSRi9GOUYvRixGLUYvKipGLkYvRjlGL0Y7Ri9GP0YtRi8qKkZYRi9GO0YvRjpGL0YsRi1GLUkjUDBHRilGLSomRj9GLUY5Ri9GLw==

u5form:=collect((factor( De*S(x)*S(1/x)-solve(u3form,Sneg(1/x))-P0-subs(x=1/x,solve(u3form,Sneg(1/x)) ))),[De,F0,F1,F2],expand);

LCwqJiwsKiQpLUkiU0c2IjYjKiRJInhHRikhIiIiIiMiIiJGLSomRidGLy1GKDYjRixGL0YvKiQpRjFGLkYvRi0qJkYsRi9GMUYvRi0qJkYsRi1GJ0YvRi1GL0kjRGVHRilGL0YvKiZGLkYvSSNGMEdGKUYvRi8qJiwmRixGL0YrRi9GL0kjRjFHRilGL0YvKiYsJiokKUYsRi5GL0YvKiRGQEYtRi9GL0kjRjJHRilGL0YvSSNQMEdGKUYt

factor(subs(F0=F0expr,F1=F1expr,F2=F2expr,u5form)/u5);

IiIi

Let us check

n:=15: expand(series(subs(De=Delta,S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)),R(0)=subs(x=0,Rser(n)),D(R)(0)=coeff(Rser(n),x,1) ,(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0), u5),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM6

<Text-field style="Heading 3" layout="Heading 3">Is the constant term in TC divisible by Delta, or at least by one of its factors ? no</Text-field>

tc:=subs(De=0,u5);

LDwqKiIiIyIiIilJInRHNiJGJEYlLUkiUkdGKDYjIiIhRiUpSSJ4R0YoRiRGJUYlKiojRiUiIiVGJS0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkiREdGNkYkNiNJIlNHRihGK0YlRiZGJUYuRiVGJSooRiRGJUYmRiUpRilGJEYlRiUqKiMiIiRGJEYlLS1GOjYjRipGK0YlRiZGJUYuRiVGJSomRiZGJUYtRiVGJSooRiRGJUYmRiVGKUYlRiUqKiMiIiZGJEYlRilGJUYnRiVGLkYlISIiKipGMEYlRjJGJUYmRiVGLkZKRiUqKkZARiVGQkYlRiZGJUYuRkpGJSoqRiRGJUYmRiVGKUYlRi1GSkYlKipGSEYlRilGJUYnRiVGLkZKRkpJI1AwR0YoRkoqJkYtRkpGJkYlRiU=

Expression of Delta in terms of s:=x+1/x

factor(subs(s=x+1/x,(1-t*(s+2))*(1-t*(s-2))-Delta));

IiIh

Deltas:=(1-t*(s+2))*(1-t*(s-2));

KiYsJiIiIkYkKiZJInRHNiJGJCwmSSJzR0YnRiQiIiNGJEYkISIiRiQsJkYkRiQqJkYmRiQsJkYpRiRGKkYrRiRGK0Yk

Expression of tc in terms of s

tcs:=collect(tc+coeff(tc,x,2)*(s^2-(x+1/x)^2),x,factor): tcs:=expand(tcs+coeff(tc,x,1)*(s-(x+1/x)));

LDQqKiIiIyIiIi1JIlJHNiI2IyIiIUYlKUkic0dGKEYkRiUpSSJ0R0YoRiRGJUYlKihGJEYlRi1GJSlGJkYkRiVGJSomRitGJUYtRiVGJSooRiRGJUYtRiVGJkYlISIiKiZGJEYlRi1GJUYzSSNQMEdGKEYzKiojRiUiIiVGJUYtRiUtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiRJIkRHRj1GJDYjSSJTR0YoRilGJUYsRiVGJSoqIyIiJEYkRiVGLUYlLS1GQTYjRidGKUYlRixGJUYlKiojIiImRiRGJUYuRiVGJkYlRixGJUYz

n:=6: expand(series(subs(R(0)=subs(x=0,Rser(n)), D(R)(0)=subs(x=0,diff(Rser(n),x)) ,(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),s=1/t-2,tcs),t,n));

KzFJInRHNiIiIiIiIiEhIiVGJSIiI0YoRiciIiQiIiYiIiUhI0dGKi1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

expand(series(subs(R(0)=subs(x=0,Rser(n)), D(R)(0)=subs(x=0,diff(Rser(n),x)) ,(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),s=1/t+2,tcs),t,n));

KzFJInRHNiIiIiIiIiEiIiVGJSIiI0YoRiciIiQiIiZGJyIjR0YqLUkiT0clKnByb3RlY3RlZEc2I0YlIiIn

What to remember from the previous sub-section

u5:=(-S(1/x)^2+S(1/x)*S(x)-S(x)^2-x*S(x)-S(1/x)/x)*De+2*t^2*R(0)*x^2+(1/4)*((D@@2)(S))(0)*t^2*x+2*t^2*R(0)^2+(3/2)*(D(R))(0)*t^2*x+t^2*x^2+2*t^2*R(0)-(5/2)*R(0)*t*x+(1/4)*((D@@2)(S))(0)*t^2/x+(3/2)*(D(R))(0)*t^2/x+2*t^2*R(0)/x^2-(5/2)*R(0)*t/x-P0+t^2/x^2;

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

u5form:=(-S(1/x)^2+S(1/x)*S(x)-S(x)^2-x*S(x)-S(1/x)/x)*De+2*F0+(x+1/x)*F1+(x^2+1/x^2)*F2-P0;

LCwqJiwsKiQpLUkiU0c2IjYjKiRJInhHRikhIiIiIiMiIiJGLSomRidGLy1GKDYjRixGL0YvKiQpRjFGLkYvRi0qJkYsRi9GMUYvRi0qJkYnRi9GLEYtRi1GL0kjRGVHRilGL0YvKiZGLkYvSSNGMEdGKUYvRi8qJiwmRixGL0YrRi9GL0kjRjFHRilGL0YvKiYsJiokKUYsRi5GL0YvKiRGQEYtRi9GL0kjRjJHRilGL0YvSSNQMEdGKUYt

F0expr:=t^2*R(0)*(R(0)+1); F1expr:=-(1/4)*t*(-6*(D(R))(0)*t-((D@@2)(S))(0)*t+10*R(0)); F2expr:=t^2*(2*R(0)+1);

KigpSSJ0RzYiIiIjIiIiLUkiUkdGJTYjIiIhRicsJkYoRidGJ0YnRic=

LCQqKCMiIiIiIiVGJUkidEc2IkYlLCgqKCIiJ0YlLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0kiUkdGKDYjIiIhRiVGJ0YlISIiKiYtLS1JI0BAR0YvNiRGLiIiIzYjSSJTR0YoRjRGJUYnRiVGNiomIiM1RiUtRjNGNEYlRiVGJUY2

KiYpSSJ0RzYiIiIjIiIiLCYqJkYmRictSSJSR0YlNiMiIiFGJ0YnRidGJ0Yn

<Text-field style="Heading 2" layout="Heading 2">An equation for M(0,x) (or S(x))</Text-field>

A multiplication that decouples terms in x and 1/x

u11:=collect(expand(u5form*(x+1/x+S(x)+S(1/x))),[De,S],expand);

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

factor(u11/subs(x=1/x,u11));

IiIi

Paper version

eqpap:=De*(S(x)^3+S(1/x)^3+(2*x+1/x)*S(x)^2+(2/x+x)*S(1/x)^2+x*(x+1/x)*S(x)+1/x*(x+1/x)*S(1/x))-(x+1/x+S(x)+S(1/x))*(2*F0-P0+(x+1/x)*F1+(x^2+1/x^2)*F2);

LCYqJkkjRGVHNiIiIiIsLiokKS1JIlNHRiU2I0kieEdGJSIiJEYmRiYqJCktRis2IyokRi0hIiJGLkYmRiYqJiwmKiYiIiNGJkYtRiZGJkYzRiZGJilGKkY4RiZGJiomLCYqJkY4RiZGLUY0RiZGLUYmRiYpRjFGOEYmRiYqKEYtRiYsJkYtRiZGM0YmRiZGKkYmRiYqKEYtRjRGP0YmRjFGJkYmRiZGJiomLCpGLUYmRjNGJkYqRiZGMUYmRiYsKiomRjhGJkkjRjBHRiVGJkYmSSNQMEdGJUY0KiZGP0YmSSNGMUdGJUYmRiYqJiwmKiQpRi1GOEYmRiYqJEZMRjRGJkYmSSNGMkdGJUYmRiZGJkY0

factor(u11/eqpap);

ISIi

Extract the non-negative part in x

First, the series in x have a negative contribution, which we must subtract

normal(subs(S(0)=0,series(subs(S(1/x)=0,De=Delta,eqpap),x,3)));

KytJInhHNiIsJEkjRjJHRiQhIiIhIiQsJEkjRjFHRiRGJyEiIywuKiYpSSJ0R0YkIiIjIiIiKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJIlNHRiQ2IyIiIUYwRjFGMSomRi5GMUYzRjFGMSomRiZGMUYzRjFGJyomRjBGMUkjRjBHRiRGMUYnRiZGJ0kjUDBHRiRGMUYnLUkiT0dGNzYjRjFGPA==

Then, the series in 1/x contibute a few non-negative terms

collect(eqpap-subs(S(1/x)=0,eqpap),[De,F0,P0,F1,F2],factor);

LCwqLC1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCgqJkYkRitGKUYrRisqJClGKSIiI0YrRitGK0YrRissJkYtRitGK0YrRitGL0YqSSNEZUdGJkYrRisqKEYwRitGJEYrSSNGMEdGJkYrRioqJkYkRitJI1AwR0YmRitGKyoqLCZGLkYrRitGK0YrRiRGK0YpRipJI0YxR0YmRitGKioqLCYqJClGKSIiJUYrRitGK0YrRitGJEYrRi9GKkkjRjJHRiZGK0Yq

normal(subs(S(0)=0,series(subs(x=1/x,De=Delta,%),x,4)));

KylJInhHNiIsKComKUkidEdGJCIiIyIiIiktLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJTR0YkNiMiIiFGKUYqRioqJkYnRipGLEYqRioqJkkjRjJHRiRGKkYsRiohIiJGOSwuKihGKUYqRitGKkYoRipGOSooRilGKkYoRipGLEYqRjkqKEYnRipGLEYqLS0tSSNAQEdGLzYkRi5GKUYyRjRGKkYqKigjRipGKUYqRidGKkY+RipGKiomRixGKkkjRjFHRiRGKkY5KihGREYqRj5GKkY4RipGOUY1LUkiT0dGMDYjRipGKg==

eqSp:=subs(S(1/x)=0,eqpap)-convert(%%%,polynom)+subs(x=1/x,convert(%,polynom));

LDoqJkkjRGVHNiIiIiIsKCokKS1JIlNHRiU2I0kieEdGJSIiJEYmRiYqJiwmKiYiIiNGJkYtRiZGJiokRi0hIiJGJkYmKUYqRjJGJkYmKihGLUYmLCZGLUYmRjNGJkYmRipGJkYmRiZGJiomLChGLUYmRjNGJkYqRiZGJiwqKiZGMkYmSSNGMEdGJUYmRiZJI1AwR0YlRjQqJkY3RiZJI0YxR0YlRiZGJiomLCYqJClGLUYyRiZGJiokRkNGNEYmRiZJI0YyR0YlRiZGJkYmRjQqJkZFRiYpRi1GLkY0RiYqJkY/RiZGQ0Y0RiYqJiwuKiYpSSJ0R0YlRjJGJiktLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRis2IyIiIUYyRiZGJiomRkxGJkZPRiZGJiomRkVGJkZPRiZGNEY7RjRGRUY0Rj1GJkYmRi1GNEY0KiYsKEZLRiZGWEYmRllGNEYmRi1GJkYmKihGMkYmRk5GJkZNRiZGNCooRjJGJkZNRiZGT0YmRjQqKEZMRiZGT0YmLS0tSSNAQEdGUjYkRlFGMkZVRlZGJkYmKigjRiZGMkYmRkxGJkZpbkYmRiYqJkZPRiZGP0YmRjQqKEZfb0YmRmluRiZGRUYmRjQ=

Let us check

n:=15: expand(series(subs(De=Delta,S(x)=Sser(n),F0=F0expr, F1=F1expr, F2=F2expr, D(S)(0)=R(0), R(0)=subs(x=0,Rser(n)), P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0), D(R)(0)=coeff(Rser(n),x,1) ,(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)),eqSp),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM6

eqSp:=subs(D(S)(0)=R(0),eqSp);

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

paper version

eqSppap:=De*(S(x)^3+(2*x+1/x)*S(x)^2+x*(x+1/x)*S(x))-(2*F0-P0+(x+1/x)*F1+(x^2+1/x^2)*F2)*(x+S(x)) -t^2*(x-1/x)*(1+R(0))^2-(1+R(0))*(F1+2*t*R(0)) ;

LCoqJkkjRGVHNiIiIiIsKCokKS1JIlNHRiU2I0kieEdGJSIiJEYmRiYqJiwmKiYiIiNGJkYtRiZGJiokRi0hIiJGJkYmKUYqRjJGJkYmKihGLUYmLCZGLUYmRjNGJkYmRipGJkYmRiZGJiomLCoqJkYyRiZJI0YwR0YlRiZGJkkjUDBHRiVGNComRjdGJkkjRjFHRiVGJkYmKiYsJiokKUYtRjJGJkYmKiRGQkY0RiZGJkkjRjJHRiVGJkYmRiYsJkYqRiZGLUYmRiZGNCooKUkidEdGJUYyRiYsJkYtRiZGM0Y0RiYpLCYtSSJSR0YlNiMiIiFGJkYmRiZGMkYmRjQqJkZLRiYsJkY+RiYqKEYyRiZGSEYmRkxGJkYmRiZGNA==

collect(factor(subs(F2=F2expr,eqSp-eqSppap)),x,factor);

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factor(subs(S(0)=0,D(S)(0)=R(0),F2=F2expr,series(subs(De=Delta,eqSppap),x,4)));

KydJInhHNiIsJCooIiIjIiIiLCYtSSJSR0YkNiMiIiFGKEYoRihGKCwmSSNGMUdGJEYoKihGJ0YoSSJ0R0YkRihGKkYoRihGKCEiIkYtLUkiT0clKnByb3RlY3RlZEc2I0YoRig=

eqSp1:=subs(F1=-2*t*R(0),F2=F2expr,F0=F0expr,eqSppap);

LCgqJkkjRGVHNiIiIiIsKCokKS1JIlNHRiU2I0kieEdGJSIiJEYmRiYqJiwmKiYiIiNGJkYtRiZGJiokRi0hIiJGJkYmKUYqRjJGJkYmKihGLUYmLCZGLUYmRjNGJkYmRipGJkYmRiZGJiomLCoqKkYyRiYpSSJ0R0YlRjJGJi1JIlJHRiU2IyIiIUYmLCZGPUYmRiZGJkYmRiZJI1AwR0YlRjQqKkYyRiZGN0YmRjxGJkY9RiZGNCooLCYqJClGLUYyRiZGJiokRkdGNEYmRiZGO0YmLCYqJkYyRiZGPUYmRiZGJkYmRiZGJkYmLCZGKkYmRi1GJkYmRjQqKEY7RiYsJkYtRiZGM0Y0RiYpRkFGMkYmRjQ=

eqSppap1:=De*(S(x)^3+(2*x+1/x)*S(x)^2+x*(x+1/x)*S(x))-(x+S(x))*(2*t^2*R0^2+2*t*(t*x^2+t/x^2-x-1/x+t)*R0-P0+t^2*(x^2+1/x^2))-t^2*(x-1/x)*(1+R0)^2;

LCgqJkkjRGVHNiIiIiIsKCokKS1JIlNHRiU2I0kieEdGJSIiJEYmRiYqJiwmKiYiIiNGJkYtRiZGJiokRi0hIiJGJkYmKUYqRjJGJkYmKihGLUYmLCZGLUYmRjNGJkYmRipGJkYmRiZGJiomLCZGKkYmRi1GJkYmLCoqKEYyRiYpSSJ0R0YlRjJGJilJI1IwR0YlRjJGJkYmKipGMkYmRj1GJiwsKiZGPUYmKUYtRjJGJkYmKiZGQ0Y0Rj1GJkYmRi1GNEYzRjRGPUYmRiZGP0YmRiZJI1AwR0YlRjQqJkY8RiYsJiokRkNGJkYmKiRGQ0Y0RiZGJkYmRiZGNCooRjxGJiwmRi1GJkYzRjRGJiksJkYmRiZGP0YmRjJGJkY0

factor(subs(R0=R(0),eqSp1-eqSppap1));

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What needs to be remembered from this subsection

eqSp1 := De*(S(x)^3+(2*x+1/x)*S(x)^2+x*(x+1/x)*S(x))-(2*t^2*R(0)*(R(0)+1)-P0-(2*(x+1/x))*t*R(0)+(x^2+1/x^2)*t^2*(2*R(0)+1))*(S(x)+x)-t^2*(x-1/x)*(R(0)+1)^2;

n:=10: normal(series(subs(De=Delta, S(x)=Sser(n),R(0)=subs(x=0,Rser(n)), P0= coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),eqSp1),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

<Text-field style="Heading 2" layout="Heading 2">The (generalized) quadratic method: algebraicity </Text-field>

eqS:=map(factor,subs(De=Delta,S(x)=Sx,R(0)=R0,eqSp1));

LCgqLiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRiohIiJGKCwqRiVGKEYsRi1GJkYoRipGLUYoSSNTeEdGJ0YoLCgqJkYvRihGKkYoRigqJEYpRihGKEYoRihGKCwmRi9GKEYqRihGKClGKiIiJEYtRigqKCw0KipGK0YoSSNSMEdGJ0YoKUYmRitGKClGKiIiJUYoRi0qKkYrRigpRjlGK0YoRjpGKEYpRihGLSomRjpGKEY7RihGLSoqRitGKEY5RihGOkYoRilGKEYtKipGK0YoRjlGKEYmRihGNEYoRigqJkkjUDBHRidGKEYpRihGKCooRitGKEY5RihGOkYoRi0qKkYrRihGOUYoRiZGKEYqRihGKCokRjpGKEYtRihGM0YoRilGLUYoKixGOkYoLCZGKEYtRipGKEYoLCZGKkYoRihGKEYoKSwmRjlGKEYoRihGK0YoRipGLUYt

Differentiating with respect to x_0

eqSS:=diff(eqS,Sx);

LCoqLCwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRiohIiJGKCwqRiVGKEYsRi1GJkYoRipGLUYoLCgqJkkjU3hHRidGKEYqRihGKCokRilGKEYoRihGKEYoLCZGMUYoRipGKEYoKUYqIiIkRi1GKCosRiRGKEYuRihGMUYoRilGLUYzRihGKCosRiRGKEYuRihGMUYoRi9GKEY0Ri1GKComLDQqKkYrRihJI1IwR0YnRigpRiZGK0YoKUYqIiIlRihGLSoqRitGKClGO0YrRihGPEYoRilGKEYtKiZGPEYoRj1GKEYtKipGK0YoRjtGKEY8RihGKUYoRi0qKkYrRihGO0YoRiZGKEY0RihGKComSSNQMEdGJ0YoRilGKEYoKihGK0YoRjtGKEY8RihGLSoqRitGKEY7RihGJkYoRipGKEYoKiRGPEYoRi1GKEYpRi1GKA==

We go back temporarily to M(0,x) to count the solutions (in x) of this equation

eqMS:=collect(subs(Sx=t*x*M(0,x),R0=t*M(0,0),eqSS*x),[M],factor);

LDwqKiIiIyIiIilJInRHNiIiIiVGJUkieEdGKEYlKS1JIk1HRig2JCIiIUYvRiRGJSEiIiosRiRGJSlGJ0YkRiUsLComRidGJSlGKkYpRiVGJSomRidGJSlGKkYkRiVGJSokKUYqIiIkRiVGMEYnRiVGKkYwRiVGKkYwRixGJUYwKi5GOkYlLCpGNkYlKihGJEYlRidGJUYqRiVGJUYnRiVGKkYwRiUsKkY2RiVGPUYwRidGJUYqRjBGJUYyRiVGKkYlKS1GLTYkRi9GKkYkRiVGJSowRiRGJUY8RiVGPkYlRidGJUYqRjAsJiomRiRGJUY3RiVGJUYlRiVGJUZARiVGJSomRjJGJSlGKiIiJkYlRiUqKEYkRiVGMkYlRjlGJUYwKihGJEYlRidGJUY1RiVGMComRjJGJUYqRiVGMCooRilGJUYnRiVGN0YlRjBGOEYlKiZJI1AwR0YoRiVGKkYlRiUqJkYkRiVGJ0YlRjBGKkYl

Introduce the divided difference (M(0,x)-M(0,0))/x, which is a series in t with polynomial coefficients in x

collect(subs(M(0,x)=M(0,0)+x*N(x),eqMS),[N,M],distributed,factor);

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

subs(t=0,%);

LCgqJClJInhHNiIiIiQiIiJGKComSSNQMEdGJkYoRiVGKEYoRiVGKA==

Since P0 is a multiple of t, we conclude that eqMS has already two roots X1 X2 with constant term +/- i. For the third root, which would have CT 0, we have to look one step further. We write x=tz

collect(subs(M(0,x)=M(0,0)+x*N(x),x=t*z,eqMS/t),[M,N],distributed,factor);

LD4qKilJInRHNiIiIiUiIiJJInpHRiZGKCwuKigiIiRGKEYkRigpRilGJ0YoRigqKCIiJ0YoKUYlIiIjRigpRilGLEYoISIiKihGL0YoRjBGKClGKUYxRihGMyomRixGKEY1RihGKComRi9GKEYpRihGM0YoRihGKCktSSJNR0YmNiQiIiFGPEYxRihGKCowRi9GKClGJSIiJkYoLCoqJkYwRihGNUYoRigqKEYxRihGJUYoRilGKEYoRilGM0YoRihGKCwqRkFGKEZCRjNGKUYzRihGKEYoRjVGKEY5RigtSSJOR0YmNiMqJkYlRihGKUYoRihGKCoqRjFGKEYlRigsMiooRjFGKClGJUYvRigpRilGP0YoRigqKEYnRihGJEYoRi1GKEYzKihGJ0YoRiRGKEYyRihGMyooRjFGKEYwRihGMkYoRigqKEY/RihGMEYoRjVGKEYzKiZGMEYoRilGKEYzRilGKEYoRjNGKEY5RihGKCouRixGKEZLRihGQEYoRkNGKEYyRigpRkRGMUYoRigqLkYxRihGMEYoRkBGKEZDRigsJiooRjFGKEYwRihGNUYoRihGKEYoRihGREYoRigqJkZLRihGTEYoRigqKEYxRihGJEYoRi1GKEYzKihGMUYoRiRGKEYyRihGMyomRjBGKEYyRihGKCooRidGKEYwRihGNUYoRjNGUUYzKiZJI1AwR0YmRihGKUYoRihGKUYoRjFGMw==

subs(t=0,%);

LCgqJkkjUDBHNiIiIiJJInpHRiVGJkYmRidGJiIiIyEiIg==

hence we also have a root X with CT 0, and first coefficient 2

Let us determine the first coefficients

with(gfun):

n:=20:expr:=normal(series(subs(M(0,x)=subs(x=0,y=x,Mser(n)),M(0,0)=subs(x=0,y=0,Mser(n)),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),eqMS),t,n+1)):

Xisol:=algeqtoseries(convert(expr,polynom),t,x,15,true);

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One root seems to be Catalan-ish

series(subs(t=t/2,op(2,Xisol)),t,16);

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expand(1/2/t*add(binomial(2*n,n)/(n+1)*(4*t^2)^(n+1),n=0..6));

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X0rac:=(1-sqrt(1-16*t^2))/4/t;

LCQqKCMiIiIiIiVGJSwmRiVGJSokKSwmKiYiIztGJSlJInRHNiIiIiNGJSEiIkYlRiUjRiVGMEYlRjFGJUYuRjFGJQ==

series(X0rac,t,20);

KzdJInRHNiIiIiMiIiIiIikiIiQiI2siIiYiJFMnIiIoIiVvciIiKiImO2cpIiM2IihXODMiIiM4IilzdTA5IiM6IipnSFYoPSIjPC1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjPg==

Let us check this conjecture

n:=20: normal(series(subs(Sx=Sser(n), R0=coeff(Rser(n),x,0),x=X0rac,P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),eqSS),t,n+1));

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algX0:=subs(X0(t)=X0,op(1,seriestoalgeq(series(X0rac,t,20),X0(t))));

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

tX0:=factor(solve(%,t));

LCQqKCMiIiIiIiNGJUkjWDBHNiJGJSwmKiQpRidGJkYlRiVGJUYlISIiRiU=

Differentiating w.r.t. x

eqSx:=collect(diff(eqS,x),[Sx,R0,P0],factor);

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<Text-field style="Heading 3" layout="Heading 3">Building an explanation for the Catalan-ish solution</Text-field>

What happens to our three equations when t=x/(1+x^2)/2 ?

factor(coeff(eqS,P0));factor(coeff(eqSS,P0));factor(coeff(eqSx,P0));

LCZJI1N4RzYiIiIiSSJ4R0YkRiU=

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We eliminate P0

r1:=collect(factor(expand(eqS-(Sx+x)*eqSS)),R0,factor);

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

r2:=collect(factor(expand(eqSS-eqSx)),R0,factor);

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

We eliminate the term R0^2

factor(r1*(x^2+1)/x^2+r2*(x-1)*(x+1)/x);

LCQqLCIiIyIiIiwoKihGJEYlSSJ0RzYiRiUpSSJ4R0YpRiRGJUYlKiZGJEYlRihGJUYlRishIiJGJSwmSSNTeEdGKUYlRitGJUYlLEQqKClGL0YkRiVGKEYlKUYrIiImRiVGLSooRi9GJUYoRiUpRisiIidGJUYtKihJI1IwR0YpRiVGKEYlRjNGJUYlKipGJEYlRjJGJUYoRiUpRisiIiRGJUYlKiZGMkYlKUYrIiIlRiVGJSooRi9GJUYoRiVGPkYlRiUqJkYvRiVGM0YlRiUqKkYkRiVGOUYlRihGJUY7RiVGLSooRjJGJUYoRiVGK0YlRi0qJkYyRiVGKkYlRiUqKEYvRiVGKEYlRipGJUYlKihGJEYlRi9GJUY7RiVGJSokRj5GJUYlKihGOUYlRihGJUYrRiVGJSomRi9GJUYoRiVGLSomRi9GJUYrRiVGJSokRipGJUYlRiVGM0YtRiU=

The complete expression

collect((eqSf-(Sx+x)*eqSSf)*(x^2+1)/x^2+(eqSSf-eqSxf)*(x-1)*(x+1)/x,[eqSf,eqSSf,eqSxf],factor);

LCgqKCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRipGJiEiIkklZXFTZkdGKEYqRioqKCwoKiZJI1N4R0YoRipGJkYqRipGMEYqKiZGKUYqRidGKkYqRipGJkYrSSZlcVNTZkdGKEYqRisqKkkmZXFTeGZHRihGKiwmRidGKkYqRitGKiwmRidGKkYqRipGKkYnRitGKw==

The explanation for X0

factor((x^2+1)*eqS-(x^2-1)*x*eqSx-(Sx*x^2+Sx+2*x)*eqSS);

Why X0 is a solution:

expr:=factor((x^2+1)*eqS-(x^2-1)*x*eqSx-(Sx*x^2+Sx+2*x)*eqSS);

pap:=-2*x*(1-2*t*(x+1/x))*(x+Sx)*(x+1/x+t*(x-1/x)^2*R0+Sx*(Sx+x+1/x)*(x+1/x-t*(x-1/x)^2));

LCQqLCIiIyIiIkkieEc2IkYlLCZGJUYlKihGJEYlSSJ0R0YnRiUsJkYmRiUqJEYmISIiRiVGJUYtRiUsJkkjU3hHRidGJUYmRiVGJSwqRiZGJUYsRiUqKEYqRiUpLCZGJkYlRixGLUYkRiVJI1IwR0YnRiVGJSooRi9GJSwoRi9GJUYmRiVGLEYlRiUsKEYmRiVGLEYlKiZGKkYlRjJGJUYtRiVGJUYlRi0=

factor(%%/%);

IiIi

fact:=x^3*expr/2;

KigsKCooIiIjIiIiSSJ0RzYiRiYpSSJ4R0YoRiVGJkYmKiZGJUYmRidGJkYmRiohIiJGJiwmSSNTeEdGKEYmRipGJkYmLEQqKClGLkYlRiZGJ0YmKUYqIiImRiZGLCooRi5GJkYnRiYpRioiIidGJkYsKihJI1IwR0YoRiZGJ0YmRjJGJkYmKipGJUYmRjFGJkYnRiYpRioiIiRGJkYmKiZGMUYmKUYqIiIlRiZGJiooRi5GJkYnRiZGPUYmRiYqJkYuRiZGMkYmRiYqKkYlRiZGOEYmRidGJkY6RiZGLCooRjFGJkYnRiZGKkYmRiwqJkYxRiZGKUYmRiYqKEYuRiZGJ0YmRilGJkYmKihGJUYmRi5GJkY6RiZGJiokRj1GJkYmKihGOEYmRidGJkYqRiZGJiomRi5GJkYnRiZGLComRi5GJkYqRiZGJiokRilGJkYmRiY=

Let us check which X_i cancels which term

normal(series(subs(x=op(1,Xisol),RootOf(_Z^2+1)=I,op(1,fact)),t)); normal(series(subs(x=op(2,Xisol),op(1,fact)),t));

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KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

normal(series(subs(Sx=Sser(10),x=op(1,Xisol),RootOf(_Z^2+1)=I,op(2,fact)),t)); normal(series(subs(Sx=Sser(10),x=op(2,Xisol),op(2,fact)),t));

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KytJInRHNiIiIiMiIiIiIzUiIiQiIyUqIiImLUkiT0clKnByb3RlY3RlZEc2I0YmIiIo

normal(series(subs(Sx=Sser(10),R0=coeff(Sser(n),x,1),x=op(1,Xisol),RootOf(_Z^2+1)=I,op(3,fact)),t)); normal(series(subs(Sx=Sser(10),R0=coeff(Sser(n),x,1),x=op(2,Xisol),op(3,fact)),t));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KylJInRHNiIiIiUiIiMiI19GJS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIic=

factor(subs(X0=x,algX0)/op(1,fact));

IiIi

Giving a name to the equation satisfied by X1 and X2

eqsupX:=op(3,fact);

LEQqKClJI1N4RzYiIiIjIiIiSSJ0R0YmRigpSSJ4R0YmIiImRighIiIqKEYlRihGKUYoKUYrIiInRihGLSooSSNSMEdGJkYoRilGKEYqRihGKCoqRidGKEYkRihGKUYoKUYrIiIkRihGKComRiRGKClGKyIiJUYoRigqKEYlRihGKUYoRjdGKEYoKiZGJUYoRipGKEYoKipGJ0YoRjJGKEYpRihGNEYoRi0qKEYkRihGKUYoRitGKEYtKiZGJEYoKUYrRidGKEYoKihGJUYoRilGKEY+RihGKCooRidGKEYlRihGNEYoRigqJEY3RihGKCooRjJGKEYpRihGK0YoRigqJkYlRihGKUYoRi0qJkYlRihGK0YoRigqJEY+RihGKA==

We now want to find equations for R0 and P0 by eliminating the X_i's and S(X_i)'s

Elimination of Sx (for x=X0). This is the discriminant with respect to Sx

e1:=numer(factor(resultant(eqS,eqSS,Sx))):nops(e1);

IiIk

op(1,e1); op(2,e1);

KiQpLCoqJkkidEc2IiIiIilJInhHRiciIiNGKEYoKihGK0YoRiZGKEYqRihGKEYmRihGKiEiIkYrRig=

KiQpLCoqJkkidEc2IiIiIilJInhHRiciIiNGKEYoKihGK0YoRiZGKEYqRighIiJGJkYoRipGLUYrRig=

algX0;

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

e2:=op(3,e1):

and then of x

e3:=numer(factor(resultant(e2,op(1,fact),x)));

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

A first relation between R0 and P0

eqR0P01:=op(1,op(3,e3)):

We now work with X1 (or X2)

e4:=numer(factor(resultant(eqS,op(3,fact),Sx))):nops(e4);

IiIn

seq(nops(op(i,e4)),i=1..nops(e4)); seq(indets(op(i,e4)),i=1..nops(e4));

NigiIiJGIyIiI0YkRiQiJC4j

Nig8I0kidEc2IjwjSSJ4R0YlRiZGJjwjSSNSMEdGJTwmSSNQMEdGJUYpRiRGJw==

e5:=op(6,e4):

e6:=numer(factor(resultant(e1,e5,x))):nops(e6);

IiIp

seq(nops(op(i,e6)),i=1..nops(e6)); seq(indets(op(i,e6)),i=1..nops(e6));

NioiIiIiIiNGJEYkRiRGJEYkRiQ=

Nio8IjwjSSJ0RzYiPCVJI1AwR0YmSSNSMEdGJkYlRidGJ0YnRic8I0Yp

Choosing the correct factor

n:=7: for i from 3 to 7 do normal(series(subs(R0=coeff(Rser(n),x,0),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),op(i,e6)),t,n+1));od;

KzVJInRHNiIiIiIiIiEhI0NGJSIkayMiIiMhJSN6IiIiJCIlOycpIiIlISZDQyQiIiYiJ29aNSIiJyEnJypvSiIiKC1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiKQ==

KzVJInRHNiIiIiIiIiEiI0NGJSIkayMiIiMiJSN6IiIiJCIlOycpIiIlIiZDQyQiIiYiJ29aNSIiJyInJypvSiIiKC1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiKQ==

Ky1JInRHNiIiIyIpIiIhIiUjZiMiIiMiJmdgJSIiJSInT1JnIiInLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiKQ==

Ky1JInRHNiIiIiIiIiEiIzciIiMiJC0iIiIlIiRvKSIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiKQ==

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiMhKQ==

op(7,e6);

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

Second relation between R0 and P0

eqR0P02:=op(1,op(7,e6));

LFgqKCIiKSIiIilJI1IwRzYiIiInRiUpSSJ0R0YoRilGJSEiIiooIiNzRiUpRiciIiZGJUYqRiVGLCooIiRTI0YlKUYnIiIlRiVGKkYlRiwqKiIjN0YlSSNQMEdGKEYlRjNGJSlGK0Y0RiVGJSooIiRnJEYlKUYnIiIkRiVGKkYlRiwqKkYuRiVGN0YlRjtGJUY4RiVGJSomRjNGJUY4RiVGJSooRjJGJSlGJyIiI0YlRipGJUYsKioiJEsiRiVGN0YlRkBGJUY4RiVGJSooIiNDRiVGO0YlRjhGJUYlKihGLkYlRidGJUYqRiVGLCoqRilGJSlGN0ZBRiVGQEYlKUYrRkFGJUYsKipGLkYlRjdGJUYnRiVGOEYlRiUqKCIjUUYlRkBGJUY4RiVGJSomRiRGJUYqRiVGLCoqIiM9RiVGSEYlRidGJUZJRiVGLCoqIiM1RiVGN0YlRkBGJUZJRiVGLCooRjZGJUY3RiVGOEYlRiUqKEZFRiVGJ0YlRjhGJUYlKihGKUYlRkhGJUZJRiVGLCoqRjZGJUY3RiVGJ0YlRklGJUYsKiRGOEYlRiUqJClGN0Y8RiVGJSooRlFGJUY3RiVGSUYlRiwqKEZBRiVGJ0YlRklGJUYsKiZGQUYlRkhGJUYsRjdGJQ==

Finally we eliminate P0 to get an equation for R0

e7:=factor(resultant(eqR0P01,eqR0P02,P0));

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

n:=7: for i from 3 to nops(e7) do normal(series(subs(R0=coeff(Rser(n),x,0),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),op(i,e7)),t,n+1));od;

Ky1JInRHNiIiIiIiIiEiIiUiIiMiI01GJyIkWyQiIictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIik=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Ky1JInRHNiIiIzsiIiEhJTs1IiIjIiY0VSMiIiUhJzNuQiIiJy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIik=

algR0:=collect(op(4,e7),R0,factor);

LCwqKCImJG8+IiIiKUkjUjBHNiIiIiVGJSlJInRHRigiIidGJUYlKioiJSg9I0YlKUYrRilGJSwmKiYiIz9GJSlGKyIiI0YlRiVGJSEiIkYlKUYnIiIkRiVGJSosIiMiKUYlRjNGJSwmKiYiIzZGJUYzRiVGJUYlRjVGJSwmKiYiI1FGJUYzRiVGJUYlRjVGJSlGJ0Y0RiVGJSooLCYqJiIjIypGJUYzRiVGJUYlRjVGJSlGOkY0RiVGJ0YlRiUqJkYzRiUsKComIiVKOEYlRi9GJUYlKiYiJDIiRiVGM0YlRjVGJUYlRiVGJQ==

And now we eliminate R0 to get an equation for P0

e8:=factor(resultant(eqR0P01,eqR0P02,R0));

LCQqLCIjayIiIilJInRHNiIiI0lGJSxOKiYiKE9GNSJGJSlGJyIjOUYlRiUqKCIpV0AnKkhGJUkjUDBHRihGJSlGJyIjN0YlRiUqKCIqZ1Y8dyNGJSlGMSIiI0YlKUYnIiM1RiVGJSomIilHNHhgRiVGMkYlISIiKigiKmthZDAqRiUpRjEiIiRGJSlGJyIiKUYlRiUqKCIqKUd0N1dGJUYxRiVGOEYlRjwqKCIqKls/dVFGJSlGMSIiJUYlKUYnIiInRiVGJSooIipTRlRZKUYlRjZGJUZBRiVGPComIignKm9HJUYlRjhGJUYlKigiKikqcEpnJEYlRj9GJUZJRiVGPCooIilLOUUoKUYlRjFGJUZBRiVGJSooIilbPFZqRiVGNkYlRklGJUYlKiYiJlMoZUYlRkFGJUY8KigiKFknKT0kRiVGP0YlKUYnRkhGJUY8KigiKEFUdiVGJUYxRiVGSUYlRjwqKCIlWygpRiVGNkYlRllGJUYlKiYiJF4oRiVGSUYlRiUqKCImZ1snRiVGMUYlRllGJUYlKihGZ25GJUY2RiUpRidGN0YlRiUqJkZCRiVGWUYlRiUqKCIkKG9GJUYxRiVGXW9GJUY8KiZGQkYlRjFGJUY8RiUpLFIqJiIpaykzcidGJSlGJyIjO0YlRiUqKEZlb0YlRjFGJUYtRiVGJSooIik7c3g7RiVGNkYlRjJGJUYlKiYiKjtDXEUjRiVGLUYlRiUqKCIoMycpUSlGJUYxRiVGMkYlRjwqKCIoYzlIJ0YlRjZGJUY4RiVGJSomIipHSDQhPUYlRjJGJUYlKigiKmtrIWVMRiVGMUYlRjhGJUY8KigiJ1NJKSpGJUY2RiVGQUYlRiUqJiIpS3NTaEYlRjhGJUYlKigiKjtPNUwiRiVGMUYlRkFGJUYlKigiJj8+KUYlRjZGJUZJRiVGJSomIikrUy08RiVGQUYlRjwqKCIpI3pwViJGJUYxRiVGSUYlRjwqKCIlU1FGJUY2RiVGWUYlRiUqJiIoY0lwIkYlRklGJUYlKigiJycqR15GJUYxRiVGWUYlRiUqKCIjJypGJUY2RiVGXW9GJUYlKiYiJmNMJ0YlRllGJUY8KigiJSE9JkYlRjFGJUZdb0YlRjwqJEY2RiVGJSomIiU/NUYlRl1vRiVGJSomRkpGJUYxRiVGJSIiKkYlRjdGJSksKComRjdGJUZdb0YlRiVGMUYlRiVGPEZIRiVGJQ==

n:=7: for i from 3 to nops(e8) do normal(series(subs(R0=coeff(Rser(n),x,0),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),op(i,e8)),t,n+1));od;

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Ky1JInRHNiIiIyIpIiIhIiZnJD0iIiMhJisqKioiIiUhKW84JikpKiIiJy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIik=

Ky1JInRHNiIiIiIiIiEhIikiIiMiIz8iIiUhI1MiIictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIik=

algP0:=collect(op(3,e8),P0,factor);

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

The equations for R0 and P0

algR0 := 19683*R0^4*t^6+2187*t^4*(20*t^2-1)*R0^3+81*t^2*(11*t^2-1)*(38*t^2-1)*R0^2+(92*t^2-1)*(11*t^2-1)^2*R0+t^2*(1331*t^4-107*t^2+1);

algP0 := 387420489*P0^4*t^6+3188646*t^4*(284*t^4-113*t^2-1)*P0^3+8748*t^2*(31570*t^8-96755*t^6+7251*t^4+t^2+1)*P0^2+(29962144*t^12-441273288*t^10+87261432*t^8-4754122*t^6+64860*t^4-687*t^2-8)*P0+t^4*(1102736*t^10-53770928*t^8+4286896*t^6-58740*t^4+751*t^2+8);

<Text-field style="Heading 2" layout="Heading 2">Parametrization for R0 and P0</Text-field>

We use the parametrization given by van Hoeij for Gessel walks (denoted by Tm, m like 'Mark')

t2Tm:=(Tm-1)/256/Tm^3*(Tm+3)^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpRigiIiRGKiksJkYoRiVGLEYlRixGJUYl

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(5);Tmser(6);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

The equation satisfied by R0 factors

res:=numer(factor(subs(t=sqrt(t2Tm),algR0)));

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

n:=6:normal(series(subs(Tm=Tmser(n),R0=coeff(Rser(n),x,0),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),R0=coeff(Rser(n),x,0),op(2,res)),t,n+1));

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KytJInRHNiIhJ1dARSIiISIpa09CQCIiIyIqXzZDayMiIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

R0Tm:=factor(solve(op(1,res),R0));

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

The equation for P0 factors as well

res:=numer(factor(subs(t=sqrt(t2Tm),algP0)));

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

n:=6:normal(series(subs(Tm=Tmser(n),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIhLi82XlkhKVIlIiIhITBLc1NgdTkibyIiIyExP04/S3gjcChIIiIlLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiJw==

P0Tm:=factor(solve(op(1,res),P0));

LCQqLCMiIiIiJEciRiUsLiokKUkjVG1HNiIiIiZGJUYlKiYiIiRGJSlGKiIiJUYlRiUqJiIjUUYlKUYqRi5GJSEiIiomIiQxIkYlKUYqIiIjRiVGNComIiRKJEYlRipGJUY0IiNURjRGJSksJkYqRiVGJUY0RjhGJSksJkYqRiVGLkYlRi5GNEYzRjRGNA==

Results of this subsection

t2Tm:=(1/256)*(Tm-1)*(Tm+3)^3/Tm^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(10);

Ky9JInRHNiIiIiIiIiEiIiUiIiMiI09GJyIkJ1IiIiciJSl5JSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

Later we will deal with Zm=sqrt(Tm)

series(sqrt(%),t,10);

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algTm:=numer(t^2-t2Tm);

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

R0Tm := -(Tm-1)*(Tm^2-6*Tm-11)/(Tm+3)^3; P0Tm := -(1/128)*(Tm^5+3*Tm^4-38*Tm^3-106*Tm^2-331*Tm-41)*(Tm-1)^2/((Tm+3)^3*Tm^3);

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

LCQqLCMiIiIiJEciRiUsLiokKUkjVG1HNiIiIiZGJUYlKiYiIiRGJSlGKiIiJUYlRiUqJiIjUUYlKUYqRi5GJSEiIiomIiQxIkYlKUYqIiIjRiVGNComIiRKJEYlRipGJUY0IiNURjRGJSksJkYqRiVGJUY0RjhGJSksJkYqRiVGLkYlRi5GNEYzRjRGNA==

<Text-field style="Heading 2" layout="Heading 2">Back to S(x); parametrization</Text-field>

eqS:=subs(De=Delta,R(0)=R0,S(x)=Sx,eqSp1);

LCgqKCwmIiIiRiUqKEkidEc2IkYlLCZJInhHRihGJUYlRiVGJSwmRiVGJSokRiohIiJGJUYlRi1GJSwmRiVGJSooRidGJSwmRiVGJUYqRi1GJSwmRiVGJUYsRi1GJUYlRiUsKCokKUkjU3hHRigiIiRGJUYlKiYsJiomIiIjRiVGKkYlRiVGLEYlRiUpRjVGOkYlRiUqKEYqRiUsJkYqRiVGLEYlRiVGNUYlRiVGJUYlKiYsKioqRjpGJSlGJ0Y6RiVJI1IwR0YoRiUsJkZCRiVGJUYlRiVGJUkjUDBHRihGLSoqRjpGJUY9RiVGJ0YlRkJGJUYtKigsJiokKUYqRjpGJUYlKiRGSUYtRiVGJUZBRiUsJiomRjpGJUZCRiVGJUYlRiVGJUYlRiUsJkY1RiVGKkYlRiVGLSooRkFGJSwmRipGJUYsRi1GJSlGQ0Y6RiVGLQ==

An equation between S(x), t, x and Tm

algSTm:=collect(numer(factor(subs(R0=R0Tm,P0=P0Tm,eqS))),Sx,factor):

We can decrease the degree in t using the expression of t^2 in terms of Tm

algSTmsmall:=collect(op(3,factor(rem(algSTm,numer(t^2-t2Tm),t))),[Sx,Tm],factor);

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n:=10: normal(series(subs(Tm=Tmser(n),Sx=Sser(n),algSTmsmall),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Let us check that S(x) has degree 12 over Q(x,t)

res:=numer(factor(resultant(algSTmsmall,algTm,Tm))): nops(%);

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degree(res,Sx);

IiM3

S(xt)/t is even in t

n:=10: series(subs(x=x*t,Sser(n+1)/t),t,n+1);

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series(t*x*subs(x=0,y=x*t,Mser(n)),t,10);

Ky1JInRHNiJJInhHRiQiIiMqJkYlIiIiLCYiIihGKComRiZGKEYlRihGKEYoIiIlKiZGJUYoLCgqJiIiJEYoKUYlRiZGKEYoKiYiI0FGKEYlRihGKCIjbEYoRigiIicqJkYlRigsKiomRixGKClGJUYwRihGKComIiNbRihGMUYoRigqJiIkXSNGKEYlRihGKCIkI3BGKEYoIiIpLUkiT0clKnByb3RlY3RlZEc2I0YoIiM1

An equation for S(xt)/t over Q(x,Tm)

res:=numer(factor(subs(x=x*t,Sx=t*Sxtst,t=sqrt(t2Tm),algSTmsmall))): nops(res);

IiIk

op(1,res); op(2,res);

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nops(op(3,res));

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This is the equation satisfied by S(xt)/t over Q(x,Tm)

algSxtstTm:=op(3,res):

Parametrization

xTmUm:=factor(solve(16*Tm^2*(Um^2-Tm)-x*(-2*Tm+Um+Um*Tm)*(-9*Tm+Um^2+8*Tm*Um+Tm^2-Tm*Um^2),x));

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Expansion of Um in terms of (Tm-1) (recall that Tm has constant term 1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

KzNJJFRtMUc2IiIiIiIiISNGJSIiI0YlIyEiIiIiKUYoLCYqJiNGJSIjS0YlSSJ4R0YkRiVGJSNGJSIjO0YlIiIkLCgqJiMiIigiJEciRiVGMEYlRiojIiImRjhGKiomI0YlRjhGJSlGMEYoRiVGJSIiJSwqKiYjIiNQIiQ3JkYlRjBGJUYlI0Y3IiRjI0YlKiYjIiM2RkNGJUY9RiVGKiomI0YlRkNGJSlGMEYzRiVGJUY6LCwjIiNAIiVDNUYqKiYjRkhGOEYlRjBGJUYqKiYjIiMiKSIlWz9GJUY9RiVGJSomI0Y3Rk9GJUZLRiVGKiomI0YlRlVGJSlGMEY+RiVGJSIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY3

res:=numer(factor(subs(x=xTmUm,algSxtstTm))):nops(%);indets(%%);

IiIl

PCVJJlN4dHN0RzYiSSNUbUdGJEkjVW1HRiQ=

op(1,res);op(2,res);

ISh3Jls1

KiQpSSNUbUc2IiIiKSIiIg==

degree(op(3,res),Sxtst);degree(op(4,res),Sxtst);

IiIi

IiIj

One has to go far to check that the quadratic term does not vanish

n:=10: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Sxtst=subs(x=x*t,Sser(n)/t),op(3,res)),t,n+1));

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n:=37: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Sxtst=subs(x=x*t,Sser(n)/t),op(4,res)),t,n+1));

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SxtstTmUm:=factor(solve(op(3,res),Sxtst));

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

Let's check

n:=18: normal(series(subs(Um=convert(UmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),SxtstTmUm),t,n+1));

Ky9JInRHNiJJInhHRiQiIiMsJiomRiYiIiIpRiVGJkYpRikqJiIiKEYpRiVGKUYpIiIlLCgqJiIiJEYpKUYlRjBGKUYpKiYiI0FGKUYqRilGKSomIiNsRilGJUYpRikiIicsKiomRi1GKSlGJUYtRilGKSomIiNbRilGMUYpRikqJiIkXSNGKUYqRilGKSomIiQjcEYpRiVGKUYpIiIpLCwqJiIiJkYpKUYlRkNGKUYpKiYiIykpRilGOUYpRikqJiIkbSdGKUYxRilGKSomIiUhKkhGKUYqRilGKSomIiUiKnpGKUYlRilGKSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGKSIjNw==

normal(series(subs(x=x*t,Sser(10)/t),t,10));

Expression of tM(0,xt)= S(xt)/t/x

M0xtftTmUm:=factor(SxtstTmUm/xTmUm);

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

The big term in the numerator

op(4,M0xtftTmUm);

LEQqJilJI1RtRzYiIiIkIiIiKUkjVW1HRiYiIiVGKEYoKigiIiNGKClGJUYrRigpRipGLUYoISIiKigiIztGKEYkRigpRipGJ0YoRjAqJilGJUYtRihGKUYoRjAqJClGJSIiJkYoRigqKCIjS0YoRi5GKEYqRihGKCooIiMhKkYoRiRGKEYvRihGKComRiVGKEYpRihGMComIiN0RihGLkYoRjAqKCIjJypGKEYkRihGKkYoRjAqKCIjIylGKEY1RihGL0YoRigqKEYyRihGJUYoRjNGKEYoKiRGKUYoRigqJiIkeCJGKEYkRihGMCooIiQjPkYoRjVGKEYqRihGMCooIiNVRihGJUYoRi9GKEYwKiYiJE4iRihGNUYoRjA=

collect(%,[Tm],factor);

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collect(%,[Um],factor);

LCwqKCwmSSNUbUc2IiIiIkYnRidGJyksJkYlRidGJyEiIiIiI0YnKUkjVW1HRiYiIiVGJ0YnKiwiIztGJ0YlRidGKUYnRiRGJylGLSIiJEYnRioqKkYrRidGJUYnLCoqJClGJUYyRidGJyomIiNYRicpRiVGK0YnRioqJiIjVEYnRiVGJ0YqIiNARidGJylGLUYrRidGKioqIiNLRidGOUYnLCgqJEY5RidGJyomRjJGJ0YlRidGKiIiJ0YqRidGLUYnRicqJkY5RicsKkY1RicqJiIjdEYnRjlGJ0YqKiYiJHgiRidGJUYnRioiJE4iRipGJ0Yn

The big term in the denominator

1/op(6,M0xtftTmUm);

LDYqJilJI1RtRzYiIiIjIiIiKUkjVW1HRiYiIiVGKEYoKihGJ0YoKUYlIiIkRigpRipGJ0YoISIiKiQpRiVGK0YoRigqKCIjO0YoRi1GKEYqRihGKCooIiNDRihGJEYoRi9GKEYwKiRGKUYoRjAqJkY2RihGLUYoRjAqKCIjW0YoRiRGKEYqRihGKCooIiInRihGJUYoRi9GKEYwKiYiIipGKEYkRihGMA==

collect(%,[Tm],factor);

LCwqJClJI1RtRzYiIiIlIiIiRigqKiIiI0YoLCZJI1VtR0YmRihGKiEiIkYoLCZGLEYoIiInRi1GKClGJSIiJEYoRi0qKCwmRixGKEYxRi1GKCwqKiQpRixGMUYoRigqJkYxRigpRixGKkYoRigqJiIjOkYoRixGKEYtRjFGKEYoKUYlRipGKEYoKihGL0YoRiVGKEY4RihGLSokKUYsRidGKEYt

collect(%,[Um],factor);

LCoqKCwmSSNUbUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicpSSNVbUdGJiIiJUYnRicqKiIiI0YnRiVGJywoKiQpRiVGLkYnRicqJiIjN0YnRiVGJ0YnIiIkRidGJylGK0YuRidGKCoqIiM7RidGMUYnLCZGJUYnRjRGJ0YnRitGJ0YnKiZGMUYnLChGMEYnKiYiI0NGJ0YlRidGKCIiKkYoRidGJw==

xTmUm:=-16*Tm^2*(-Um^2+Tm)/((Tm*Um-2*Tm+Um)*(-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm));

LCQqLCIjOyIiIilJI1RtRzYiIiIjRiUsJiokKUkjVW1HRihGKUYlISIiRidGJUYlLCgqJkYnRiVGLUYlRiUqJkYpRiVGJ0YlRi5GLUYlRi4sLComRidGJUYsRiVGLiokRiZGJUYlKigiIilGJUYnRiVGLUYlRiVGK0YlKiYiIipGJUYnRiVGLkYuRi4=

Expansion of Um in terms of (Tm-1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

KzNJJFRtMUc2IiIiIiIiISNGJSIiI0YlIyEiIiIiKUYoLCYqJiNGJSIjS0YlSSJ4R0YkRiVGJSNGJSIjO0YlIiIkLCgqJiMiIigiJEciRiVGMEYlRiojIiImRjhGKiomI0YlRjhGJSlGMEYoRiVGJSIiJSwqKiYjIiNQIiQ3JkYlRjBGJUYlI0Y3IiRjI0YlKiYjIiM2RkNGJUY9RiVGKiomI0YlRkNGJSlGMEYzRiVGJUY6LCwqJiNGNyIlQzVGJUZLRiVGKiomIyIjIikiJVs/RiVGPUYlRiUqJiNGJUZTRiUpRjBGPkYlRiUjIiNARk9GKiomI0ZIRjhGJUYwRiVGKiIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY3

normal(series(subs(Tm1=Tmser(10)-1,%),t,11));

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSwmIiRtIkYlKiZGJ0YlSSJ4R0YkRiVGJSIiJywoKiZGJ0YlKUYtRidGJUYlKiYiI1NGJUYtRiVGJSIlTT5GJSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

SxtstTmUm := 16*Tm*(-Um^2+Tm)*(Tm*Um-2*Tm+Um)*(Tm^3*Um^4-2*Tm^4*Um^2-16*Tm^3*Um^3-Tm^2*Um^4+Tm^5+32*Tm^4*Um+90*Tm^3*Um^2-Tm*Um^4-73*Tm^4-96*Tm^3*Um+82*Tm^2*Um^2+16*Tm*Um^3+Um^4-177*Tm^3-192*Tm^2*Um-42*Tm*Um^2-135*Tm^2)/((Tm+3)^3*(-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm)*(Tm^2*Um^4-2*Tm^3*Um^2+Tm^4+16*Tm^3*Um-24*Tm^2*Um^2-Um^4-24*Tm^3+48*Tm^2*Um-6*Tm*Um^2-9*Tm^2));

M0xtftTmUm:=factor(SxtstTmUm/xTmUm);

algSTmsmall := (x*(x+1)^2*(x-1)^2*Tm^7-x*(512*t*x^3-17*x^4+512*t*x-222*x^2-17)*Tm^6-9*x*(512*t*x^3-13*x^4+512*t*x-230*x^2-13)*Tm^5-27*x*(512*t*x^3-15*x^4+512*t*x-226*x^2-15)*Tm^4-27*x*(512*t*x^3-25*x^4+512*t*x-206*x^2-25)*Tm^3+243*x*(x+1)^2*(x-1)^2*Tm^2-729*x*(x+1)^2*(x-1)^2*Tm-729*x*(x+1)^2*(x-1)^2)*Sx^3+((2*x^2+1)*(x+1)^2*(x-1)^2*Tm^7-(2*x^2+1)*(512*t*x^3-17*x^4+512*t*x-222*x^2-17)*Tm^6-(9*(2*x^2+1))*(512*t*x^3-13*x^4+512*t*x-230*x^2-13)*Tm^5-(27*(2*x^2+1))*(512*t*x^3-15*x^4+512*t*x-226*x^2-15)*Tm^4-(27*(2*x^2+1))*(512*t*x^3-25*x^4+512*t*x-206*x^2-25)*Tm^3+(243*(2*x^2+1))*(x+1)^2*(x-1)^2*Tm^2-(729*(2*x^2+1))*(x+1)^2*(x-1)^2*Tm-(729*(2*x^2+1))*(x+1)^2*(x-1)^2)*Sx^2+(x*(x^2+2)*(x+1)^2*(x-1)^2*Tm^7-x*(512*t*x^5-17*x^6+1536*t*x^3-224*x^4+1024*t*x-269*x^2-2)*Tm^6-x*(4608*t*x^5-117*x^6+5632*t*x^3-1984*x^4+1024*t*x-2081*x^2+86)*Tm^5-x*(13824*t*x^5-405*x^6+25088*t*x^3-5792*x^4+11264*t*x-6145*x^2+310)*Tm^4-x*(13824*t*x^5-675*x^6+33280*t*x^3-5504*x^4+19456*t*x-6935*x^2+58)*Tm^3+3*x*(81*x^6+96*x^4-179*x^2+258)*Tm^2-9*x*(81*x^6-192*x^4-115*x^2-30)*Tm-27*x*(27*x^2+22)*(x+1)^2*(x-1)^2)*Sx+(x+1)^2*(x-1)^2*x^2*Tm^7-x^2*(512*t*x^3+15*x^4+512*t*x-30*x^2+15)*Tm^6+x^2*(3584*t*x^3-203*x^4+3584*t*x-362*x^2+53)*Tm^5+x^2*(2560*t*x^3-715*x^4+2560*t*x-1130*x^2+53)*Tm^4-x^2*(5632*t*x^3+733*x^4+5632*t*x-186*x^2+221)*Tm^3+x^2*(531*x^4+218*x^2+19)*Tm^2+3*x^2*(37*x^2+11)*(9*x^2+7)*Tm+x^2*(x-1)*(x+1)*(135*x^2+121);

<Text-field style="Heading 2" layout="Heading 2">Determination of M(x,0) (or R(x)) and parametrization</Text-field>

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGL0YwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjBGL0YqRjZGMComRjZGKilGKUYsRjBGMA==

We first find an equation for DD:=S(x)-2S(1/x)-1/x

indets(algSTmsmall);

PCZJI1N4RzYiSSNUbUdGJEkidEdGJEkieEdGJA==

res:=numer(factor(resultant(subs(Sx=DD+2*S1x+1/x,algSTmsmall),subs(Sx=S1x,x=1/x,algSTmsmall),S1x))): nops(res);

IiIm

seq(nops(op(i,res)),i=1..nops(res));

NiciIiIiIiNGJCIkXyIiJWc9

seq(degree(op(i,res),DD),i=1..nops(res));

NiciIiFGI0YjIiIkIiIn

Checking which factor vanishes

n:=1: normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))-1/x,Tm=Tmser(n),op(4,res)),t,n+1));

KydJInRHNiIsKComIiZvRiQiIiIpSSJ4R0YkIiIpRighIiIqJiImLyQpKkYoKUYqIiIlRihGKComIiZPYidGKClGKiIiI0YoRigiIiEtSSJPRyUqcHJvdGVjdGVkRzYjRihGKA==

n:=10:normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))-1/x,Tm=Tmser(n),op(5,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

We decrease the degree in t

degree(op(5,res),t);

IiIk

res1:=factor(rem(op(5,res),numer(t^2-t2Tm),t)):nops(%);

IiIj

op(1,res);

ISIi

algDDTm:=op(2,res1):

Hence DD has degree 6 over Q(x,T,t)

degree(algDDTm,DD);

IiIn

But is in fact bicubic

seq(coeff(algDDTm,DD,2*i+1),i=0..2);

NiUiIiFGI0Yj

DD has degree 24 over Q(x,t)

res:=factor(resultant(algDDTm,numer(t^2-t2Tm),Tm)):nops(res);

IiIk

seq(nops(op(i,res)),i=1..nops(res));seq(degree(op(i,res),DD),i=1..nops(res));

NiUiIiIiIiMiJSFcKQ==

NiUiIiFGIyIjQw==

Going back to R(x)

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGL0YwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjBGL0YqRjZGMComRjZGKilGKUYsRjBGMA==

res:=numer(factor(subs(DD=(3*t*Rx-1/x+t+t/x^2)/sqrt(Delta),algDDTm))): nops(%);

IiIj

op(1,res);nops(op(2,res));

KiQpSSJ4RzYiIiIjIiIi

IiV2dg==

degree(op(2,res),Rx);degree(op(2,res),t);

IiIn

IiIo

We lower the degree in t by taking a remainder

numer(factor(rem(op(2,res),numer(t^2-t2Tm),t))): nops(%);

IiUleSU=

algRTm:=%%:

indets(algRTm);

PCZJI1J4RzYiSSNUbUdGJEkidEdGJEkieEdGJA==

Equation of R(x) over Q(t,x,T)

res:=numer(factor(resultant(algRTm,numer(t^2-t2Tm),Tm))): nops(%);

IiIm

seq(nops(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rx),i=1..nops(res));

NiciIiIiIiNGJEYkIiV6Iyo=

NiciIiFGI0YjRiMiI0M=

Equation satisfied by R(x) over Q(t,x)

algR:=op(5,res):

n:=10:normal(series(subs(Rx=Rser(n),algR),t,n+1));

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Parity

subs(x=x*t,Rser(3));

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res:=factor(subs(x=x*t,t=sqrt(t2Tm),Rx=Rxt,algRTm)): nops(res);

IiIn

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNUbUc2IkYkPCRGJUkieEdGJjwlSSRSeHRHRiZGJUYoRiQ=

algRxtTm:=collect(op(5,res),Rxt,factor):

degree(algRxtTm,Rxt);

IiIn

Parametrisation: we first introduce Zm=sqrt(Tm)

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

Zmser(5);

KytJInRHNiIiIiIiIiEiIiNGJyIjOyIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

The degree over Q(Zm,x) is only three

res:=factor(subs(Tm=Zm^2,algRxtTm)): nops(res);

IiIj

seq(indets(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rxt),i=1..nops(res));

NiQ8JUkkUnh0RzYiSSNabUdGJUkieEdGJUYj

NiQiIiRGIw==

Choosing the correct factor

n:=6: normal(series(subs(Zm=Zmser(n),Rxt=subs(x=x*t,Rser(n)),op(1,res)),t,n+1));

KytJInRHNiIsKComIi0pKVEiZXZcJiIiIilJInhHRiQiIiRGKEYoKiYiLms7V24jXDtGKClGKiIiI0YoISIiKiYiLnd4aTYmKjQiRihGKkYoRigiIiEsLComRidGKClGKiIiJkYoRigqJiIuR0wpW2ApSCRGKClGKiIiJUYoRjAqJiIvL2I/UGUpPSRGKEYpRihGKComIi9HbCdwIT53JilGKEYuRihGMComIi9fVmsvWTxkRihGKkYoRihGLywuKiZGLUYoKUYqIiInRihGMComRj1GKEY2RihGKComIjAzJyk9UEdzdCJGKEY6RihGMComIjFXUHFbQlM/NkYoRilGKEYoKiYiMV8oeVpZVE0kR0YoRi5GKEYwKiYiMW8iPmwoNCcqKSk9RihGKkYoRihGOy1JIk9HJSpwcm90ZWN0ZWRHNiNGKEZF

n:=10: normal(series(subs(Zm=Zmser(n),Rxt=subs(x=x*t,Rser(n)),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

algRxtZm:=collect(op(2,res),Rxt,factor):

And finally we parametrize x by Um

res:=numer(factor(subs(x=xTmUm,Tm=Zm^2,algRxtZm))): nops(res);

IiIl

seq(indets(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rxt),i=1..nops(res));

NiY8IjwjSSNabUc2IjwlSSRSeHRHRiZJI1VtR0YmRiVGJw==

NiYiIiFGIyIiIiIiIw==

One has to go far to determine which term vanishes

n:=10: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rxt=subs(x=x*t,Rser(n)),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=44: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rxt=subs(x=x*t,Rser(n)),op(4,res)),t,n+1));

KydJInRHNiIsKComIi1XcCF6KFtGIiIiKUkieEdGJCIiI0YoRigqJiItKSlRImV2XCZGKEYqRighIiJGJ0YoIiNVLUkiT0clKnByb3RlY3RlZEc2I0YoIiNX

Rational expression of R(xt) in terms of Zm and Um

RxtZmUm:=factor(solve(op(3,res),Rxt));

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

Mxt0ftZmUm:=RxtZmUm:

denom(Mxt0ftZmUm);

LCQqMCIiJCIiIilJI1ptRzYiIiIjRiUsJkYnRiVGJSEiIkYlKSwmKiRGJkYlRiVGJEYlRiRGJSksKiomSSNVbUdGKEYlRidGJUYlRi5GJUYyRiUqJkYkRiVGJ0YlRitGKUYlKSwmRjJGJUYnRiVGKUYlKSwqRjFGJUYuRitGMkYrRjNGK0YpRiVGJQ==

factor(subs(Zm=sqrt(Tm),expand((Um*Zm+Zm^2+Um-3*Zm)*(Um*Zm-Zm^2-Um-3*Zm))));

LCwqJkkjVG1HNiIiIiIpSSNVbUdGJSIiI0YmRiYqJClGJEYpRiYhIiIqKCIiKUYmRiRGJkYoRiZGLCokRidGJkYsKiYiIipGJkYkRiZGJg==

num:=collect(numer(Mxt0ftZmUm),Um,factor);

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

collect(%,Zm,factor);

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

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

RxtZmUm := factor(-(1/3)*(Um^8*Zm^11+2*Um^7*Zm^12-2*Um^5*Zm^14-2*Um^4*Zm^15-2*Um^3*Zm^16+2*Um*Zm^18+Zm^19-Um^8*Zm^10-22*Um^7*Zm^11-40*Um^6*Zm^12-26*Um^5*Zm^13-14*Um^4*Zm^14+22*Um^3*Zm^15+56*Um^2*Zm^16+26*Um*Zm^17-Zm^18-Um^8*Zm^9+18*Um^7*Zm^10+180*Um^6*Zm^11+182*Um^5*Zm^12+246*Um^4*Zm^13+414*Um^3*Zm^14-4*Um^2*Zm^15-230*Um*Zm^16-37*Zm^17+Um^8*Zm^8-6*Um^7*Zm^9-156*Um^6*Zm^10-658*Um^5*Zm^11+570*Um^4*Zm^12-298*Um^3*Zm^13-1860*Um^2*Zm^14-190*Um*Zm^15+293*Zm^16-2*Um^8*Zm^7+4*Um^7*Zm^8+336*Um^6*Zm^9+636*Um^5*Zm^10+1508*Um^4*Zm^11-5892*Um^3*Zm^12-2736*Um^2*Zm^13+2692*Um*Zm^14+382*Zm^15+2*Um^8*Zm^6+52*Um^7*Zm^7-224*Um^6*Zm^8-2580*Um^5*Zm^9+1148*Um^4*Zm^10-4404*Um^3*Zm^11+14816*Um^2*Zm^12+7444*Um*Zm^13-894*Zm^14+2*Um^8*Zm^5-44*Um^7*Zm^6-328*Um^6*Zm^7+2236*Um^5*Zm^8+8324*Um^4*Zm^9-17012*Um^3*Zm^10+8136*Um^2*Zm^11-16668*Um*Zm^12-4614*Zm^13-2*Um^8*Zm^4+4*Um^7*Zm^5+248*Um^6*Zm^6+76*Um^5*Zm^7-8740*Um^4*Zm^8-18532*Um^3*Zm^9+45096*Um^2*Zm^10-8940*Um*Zm^11+7686*Zm^12+Um^8*Zm^3-6*Um^7*Zm^4-304*Um^6*Zm^5+70*Um^5*Zm^6+2174*Um^4*Zm^7+11142*Um^3*Zm^8+27024*Um^2*Zm^9-48582*Um*Zm^10+5409*Zm^11-Um^8*Zm^2-30*Um^7*Zm^3+232*Um^6*Zm^4+1134*Um^5*Zm^5-4302*Um^4*Zm^6-8034*Um^3*Zm^7-2808*Um^2*Zm^8-25326*Um*Zm^9+17631*Zm^10-Um^8*Zm+26*Um^7*Zm^2+116*Um^6*Zm^3-1074*Um^5*Zm^4-2010*Um^4*Zm^5+7254*Um^3*Zm^6+12636*Um^2*Zm^7-702*Um*Zm^8+9099*Zm^9+Um^8+2*Um^7*Zm-60*Um^6*Zm^2+6*Um^5*Zm^3+1098*Um^4*Zm^4+2574*Um^3*Zm^5-2052*Um^2*Zm^6-7830*Um*Zm^7-2187*Zm^8)/(Zm^2*(Zm-1)*(Zm^2+3)^3*(Um*Zm+Zm^2+Um-3*Zm)^2*(Um+Zm)^2*(Um*Zm-Zm^2-Um-3*Zm)^2));

<Text-field style="Heading 2" layout="Heading 2">Back to M(x,y) and P(x,y): degrees 72 over Q(x,y,t)</Text-field>

eqcatM;

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRiZGKEYqRihGKComRiZGKEYsRihGKComRipGKEYsRighIiJGKCwmKiZGK0YoLUkiTUdGJzYkRipGLEYoRigtRjY2JCIiIUYsRjJGKEYqRjJGLEYyRjIqJiNGKyIiJEYoRipGKEYyKipGK0YoRiZGKC1GNjYkRipGOkYoRixGMkYoKixGJkYoLCZGKkYoRihGMkYoLCZGKkYoRihGKEYoRjhGKEYqRjJGMiooRiZGKC1GNjYkRixGOkYoRixGMkYy

Two even series in t

n:=8:series(subs(x=x*t,y=y*t,t*Mser(n)),t,n+1);

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

n:=6:series(subs(x=x*t,y=y*t,Pser(n)),t,n+1);

Ky1JInRHNiIjIiIjIiIkIiIhLCgjIiM1RiciIiIqJkYlRixJInhHRiRGLEYsKiZGJUYsSSJ5R0YkRixGLEYmLC4jIiMnKUYnRiwqJiMiI0FGJ0YsRi5GLEYsKiZGNUYsRjBGLEYsKigjIiIlRidGLEYuRixGMEYsRiwqJkYlRiwpRi5GJkYsRiwqJkYlRiwpRjBGJkYsRixGOiw2IyIkJSkpRidGLComIyIkWiNGJ0YsRi5GLEYsKiZGQ0YsRjBGLEYsKigjIiNjRidGLEYuRixGMEYsRiwqJiIjOEYsRjxGLEYsKiZGSkYsRj5GLEYsKihGJkYsRjxGLEYwRixGLComRiVGLClGLkYnRixGLCooRiZGLEYuRixGPkYsRiwqJkYlRiwpRjBGJ0YsRiwiIictSSJPRyUqcHJvdGVjdGVkRzYjRiwiIik=

An equation for tM(xt,yt)

eqMxtyt:=map(Z->factor(t*Z),subs(x=x*t,y=y*t,M(t*x,t*y)=Mxtytft/t,M(t*x,0)=Mxt0ft/t,M(t*y,0)=Myt0ft/t,M(0,y*t)=M0ytft/t,eqcatM));

LCwqKiwsKigpSSJ0RzYiIiIjIiIiKUkieEdGKEYpRipJInlHRihGKkYqKihGJkYqRixGKilGLUYpRipGKiomRixGKkYtRiohIiJGLEYqRi1GKkYqLCYqJkYpRipJKE14dHl0ZnRHRihGKkYxSSdNMHl0ZnRHRihGKkYqRixGMUYtRjFGKiooI0YpIiIkRipGJkYqRixGKkYxKihGKUYqSSdNeHQwZnRHRihGKkYtRjFGKioqLCYqJkYnRipGLEYqRipGKkYxRiosJkY9RipGKkYqRipGNUYqRixGMUYxKiZJJ015dDBmdEdGKEYqRi1GMUYx

solve(eqMxtyt,Mxtytft);

LCQqKCMiIiIiIidGJSwwKiwiIiRGJUknTTB5dGZ0RzYiRiUpSSJ0R0YrIiIjRiVJInhHRitGJSlJInlHRitGLkYlRiUqKkYuRiVGLEYlKUYvRi5GJUYxRiUhIiIqKkYpRiVGKkYlRi9GJUYxRiVGNCooRilGJUYqRiVGL0YlRiUqKEYmRiVGKkYlRjFGJUYlKihGJkYlSSdNeHQwZnRHRitGJUYvRiVGJSooRilGJUknTXl0MGZ0R0YrRiVGL0YlRjRGJSwsKihGLEYlRjNGJUYxRiVGJSooRixGJUYvRiVGMEYlRiUqJkYvRiVGMUYlRjRGL0YlRjFGJUY0RiU=

The counterpart of Um for the variable y (instead of x) is denoted Umy

MxtytftZmUmUmy:=factor(subs(t=sqrt(t2Tm),x=xTmUm,y=subs(Um=Umy,xTmUm),Mxt0ft=Mxt0ftZmUm,Myt0ft=subs(Um=Umy,Mxt0ftZmUm),M0ytft=subs(Um=Umy,M0xtftTmUm),Tm=Zm^2,%)): indets(%);

PCVJI1VtRzYiSSRVbXlHRiRJI1ptR0Yk

Let us check this expression

n:=30: normal(series(subs(Um=UmserTm(n),Umy=subs(x=y,UmserTm(n)),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),MxtytftZmUmUmy),t,n+1));

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

normal(series(subs(x=x*t,y=y*t,t*Mser(10)),t,10));

normal(series(%%-%,t,10));

IiIh

Degree when x=2 and y=-2

algUmZmx2:=factor(subs(Tm=Zm^2,numer(1/2*factor(2-xTmUm))));

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algUmyZmym2:=factor(subs(Tm=Zm^2,subs(Um=Umy,numer(1/2*factor(-2-xTmUm)))));

LDQqJilJJFVteUc2IiIiJCIiIilJI1ptR0YmIiIlRihGKComRiVGKClGKiIiJ0YoISIiKigiIz1GKClGJSIiI0YoRilGKEYvKiYiIzVGKEYtRihGKCooIiNDRihGJUYoRilGKEYoKihGLkYoRjJGKClGKkYzRihGLyomRjFGKEYpRihGLyokRiRGKEYvKigiIipGKEYlRihGOUYoRig=

res:=numer(factor(resultant(numer(Mxtytft-MxtytftZmUmUmy),algUmyZmym2,Umy))): nops(res);

IiIn

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSShNeHl0eWZ0R0YmSSNVbUdGJkYl

res1:=numer(factor(resultant(op(6,res),algUmZmx2,Um))): nops(res1);

IiIn

seq(indets(op(i,res1)),i=1..nops(res1));

Nig8IjwjSSNabUc2IkYkRiRGJDwkSShNeHl0eWZ0R0YmRiU=

algMxtytftZm:=op(6,res1):

degree(algMxtytftZm,Mxtytft);

IiIq

algZm:=subs(Tm=Zm ^2,algTm);

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res2:=factor(resultant(algMxtytftZm,algZm,Zm)): nops(res2);

IiUpXCk=

degree(res2,Mxtytft);

IiNz

And now back to P(x,y). We first express P in terms of the specializations of M

PexprMsimple:=collect(subs(isolate(eqcatM,M(x,y)),subs({x=y,y=x},isolate(eqcatM,M(x,y))),PexprM),M,factor);

LCwqKiMiIiIiIiNGJSwqKiZJInRHNiJGJSlJInhHRipGJkYlRiUqKkYmRiVGKUYlRixGJUkieUdGKkYlRiVGKUYlRiwhIiJGJSwsKihGKUYlRitGJUYuRiVGJSooRilGJUYsRiUpRi5GJkYlRiUqJkYpRiVGLEYlRiUqJkYpRiVGLkYlRiUqJkYsRiVGLkYlRi9GLy1JIk1HRio2JCIiIUYsRiVGLyoqRiRGJSwqRi1GJSomRilGJUYzRiVGJUYpRiVGLkYvRiVGMEYvLUY4NiRGOkYuRiVGLyoqRiRGJUYpRiVGMEYvLUY4NiRGLEY6RiVGJSoqRiRGJUYpRiVGMEYvLUY4NiRGLkY6RiVGJSoqI0YmIiIkRiVGLEYlRi5GJUYwRi9GLw==

PxtytZmUmUmy:=factor(subs(x=x*t,y=y*t,M(t*x,0)=Mxt0ft/t,M(t*y,0)=Myt0ft/t,M(0,y*t)=M0ytft/t,M(0,x*t)=M0xtft/t,t=sqrt(t2Tm),x=xTmUm,y=subs(Um=Umy,xTmUm),Mxt0ft=Mxt0ftZmUm,Myt0ft=subs(Um=Umy,Mxt0ftZmUm),M0ytft=subs(Um=Umy,M0xtftTmUm),M0xtft=M0xtftTmUm,Tm=Zm^2,PexprMsimple)): indets(%);

PCVJI1VtRzYiSSRVbXlHRiRJI1ptR0Yk

Let us check this expression

n:=40: normal(series(subs(Um=UmserTm(n),Umy=subs(x=y,UmserTm(n)),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),PxtytZmUmUmy),t,n+1));

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

normal(series(%-subs(x=x*t,y=y*t,Pser(14)),t,14));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

Degree when x=2 and y=-2

res:=numer(factor(resultant(numer(Pxtyt-PxtytZmUmUmy),algUmyZmym2,Umy))): nops(res);

IiIn

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSZQeHR5dEdGJkkjVW1HRiZGJQ==

res1:=numer(factor(resultant(op(6,res),algUmZmx2,Um))): nops(res1);

IiIn

seq(indets(op(i,res1)),i=1..nops(res1));

Nig8IjwjSSNabUc2IkYkRiRGJDwkSSZQeHR5dEdGJkYl

algPxtytZm:=op(6,res1):

degree(algPxtytZm,Pxtyt);

IiIq

algZm:=subs(Tm=Zm ^2,algTm);

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

res2:=factor(resultant(algPxtytZm,algZm,Zm)): nops(res2);

IiYheTY=

degree(res2,Pxtyt);

IiNz

<Text-field style="Heading 2" layout="Heading 2">Walks ending at a prescribed position</Text-field>

We start with R(xt), and re-build its cubic equation over Q(x,Zm)

res:=factor(resultant(numer(RxtZmUm-Rxt),subs(Tm=Zm^2,numer(x-xTmUm)),Um)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJDYk

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSRSeHRHRiZGJUkieEdGJg==

algRxtZm:=op(6,res):indets(%);

PCVJJFJ4dEc2IkkjWm1HRiRJInhHRiQ=

collect(factor(subs(x=0,algRxtZm))/25769803776,Rxt,factor);

LCYqKClJI1ptRzYiIiNDIiIiKSwmKiQpRiUiIiNGKEYoIiIkRihGLkYoSSRSeHRHRiZGKEYoKipGJEYoLCZGJUYoRighIiJGKCwmRiVGKEYoRihGKCwoKiQpRiUiIiVGKEYoKiYiIidGKEYsRihGMiIjNkYyRihGKA==

degree(%,Rxt);

IiIi

factor(solve(%%,Rxt));

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Now for S: we first for the equation satisfied by S(xt)/t

res:=factor(resultant(numer(SxtstTmUm-Sxtst),numer(x-xTmUm),Um)): nops(res);

IiIm

seq(nops(op(i,res)),i=1..nops(res));

NiciIiIiIiNGJEYkIiQoRw==

seq(indets(op(i,res)),i=1..nops(res));

Nic8IjwjSSNUbUc2IkYkRiQ8JUkmU3h0c3RHRiZGJUkieEdGJg==

algSxtstTm:=op(5,res):indets(%);

PCVJJlN4dHN0RzYiSSNUbUdGJEkieEdGJA==

Equation for S(xt)/(xt)

algSxtstxZm:=factor(1/x^2*subs(Sxtst=x*Sxtstx,Tm=Zm^2,algSxtstTm)):

nops(%);

IiQoRw==

factor(series(subs(Sxtstx=St0+x*St1+x^2*Sh(x),algSxtstxZm),x,1));

KydJInhHNiIsJCooIik7c3g7IiIiKUkjWm1HRiQiIz1GKCksMiomSSRTdDBHRiRGKClGKiIiJ0YoRigqJEYwRihGKCooIiIqRihGL0YoKUYqIiIlRihGKComIiIoRihGNUYoISIiKigiI0ZGKEYvRigpRioiIiNGKEYoKiYiIiZGKEY8RihGOSomRjtGKEYvRihGKCIjNkYoRj1GKEYoIiIhLUkiT0clKnByb3RlY3RlZEc2I0YoRig=

factor({solve(op(1,%),St0)});

PCMsJCoqLCZJI1ptRzYiIiIiIiIiISIiIiIiLCZJI1ptRzYiIiIiIiIiIiIiIiIiLCgqJClJI1ptRzYiIiIlIiIiIiIiKiYiIiciIiIpSSNabUc2IiIiIyIiIiEiIiIjNiEiIiIiIiksJiokKUkjWm1HNiIiIiMiIiIiIiIiIiQiIiIiIiQhIiIhIiI=

St0sol:=op(%):

factor(subs(St0=St0sol,series(subs(Sxtstx=St0+x*St1+x^2*Sh(x),algSxtstxZm),x,3)));

KydJInhHNiIsJCoqIik7c3g7IiIiKUkjWm1HRiQiIzdGKCw6KiZJJFN0MUdGJEYoKUYqIiIqRihGKCooRjBGKEYuRigpRioiIihGKEYoKiYiIiVGKClGKiIiKUYoRigqJiIjO0YoRjJGKCEiIiooIiNGRihGLkYoKUYqIiImRihGKComIiNDRigpRioiIidGKEYoKiYiI0tGKEY9RihGOiooRjxGKEYuRigpRioiIiRGKEYoKiZGREYoKUYqRjVGKEYoKiZGOUYoRkZGKEY6KiZGN0YoKUYqIiIjRihGKEY1RjpGKCw6Ri1GKEYxRihGNEY6RjhGOkY7RihGP0Y6RkNGOkZFRihGSEY6RkpGOkZLRjpGNUYoRihGKEZNLUkiT0clKnByb3RlY3RlZEc2I0YoRkc=

St1solens:=factor({solve(op(1,%),St1)});

PCQsJCouIiIlIiIiLCgqJClJI1ptRzYiIiIjRiZGJiomRixGJkYqRiYhIiJGJkYuRiYpLCZGKkYmRiZGLkYsRiYpLCZGKEYmRiZGJkYsRiYpLCZGKEYmIiIkRiZGNUYuKUYqRjVGLkYuLCQqLkYlRiYsKEYoRiZGLUYmRiZGLkYmKSwmRipGJkYmRiZGLEYmRjFGJkYzRi5GNkYuRiY=

n:=10: normal(series(subs(Zm=Zmser(n),St1solens[1]),t,n+1));

Ky1JInRHNiIiIiMiIiUiI0EiIiciJF0jIiIpIiUhKkgiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

n:=10: normal(series(subs(Zm=Zmser(n),St1solens[2]),t,n+1));

Ky9JInRHNiIiIiMiIiFGJUYlIiM3IiIlIiQzIiIiJyIlTTYiIiktSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=10:expand(coeff(normal(subs(x=x*t,Sser(n))/x/t),x,1));

LCwqJiImZXMkIiIiKUkidEc2IiIjN0YlRiUqJiIlISpIRiUpRiciIzVGJUYlKiYiJF0jRiUpRiciIilGJUYlKiYiI0FGJSlGJyIiJ0YlRiUqJiIiI0YlKUYnIiIlRiVGJQ==

St1sol:=St1solens[1]:

An equation for S hat

res:=numer(1/x^3*factor(subs(Sxtstx=St0sol+x*St1sol+x^2*Sh(x),algSxtstxZm))): nops(%);

IiQkbw==

factor(subs(x=0,res));

LCQqLCIoMycpUSkiIiIpSSNabUc2IiIjPUYlLCgqJClGJyIiJUYlRiUqJkYtRiUpRiciIiNGJUYlRiUhIiJGJSksJiokRi9GJUYlRiVGJUYwRiUsQiooIiM7RiUtSSNTaEdGKDYjIiIhRiUpRiciIzdGJUYlKiYiIipGJUY8RiVGMSooIiRXIkYlRjhGJSlGJyIjNUYlRiUqJiIjJypGJSlGJyIjNkYlRiUqJiIkSSRGJUZCRiVGMSooIiRLJUYlRjhGJSlGJyIiKUYlRiUqJiIkdyZGJSlGJ0Y/RiVGJSomIiRaKEYlRkxGJUYxKihGS0YlRjhGJSlGJyIiJ0YlRiUqJiIkbyhGJSlGJyIiKEYlRiUqJiIkI1xGJUZURiVGMSomIiQjPkYlKUYnIiImRiVGJSomIiM6RiVGLEYlRjEqJkZFRiUpRiciIiRGJUYxKiYiI2FGJUYvRiVGJUZeb0YlRiVGMQ==

degree(%,Sh(0));

IiIi

<Text-field style="Heading 3" layout="Heading 3">Walks ending at abscissa (-1): the series M_{0,j} (up to ordinate 2)</Text-field>

with(gfun):

ser:=series(St0sol+x*St1sol+x^2*map(factor,op(algeqtoseries(subs(Sxtstx=St0sol+x*St1sol+x^2*Sh,algSxtstxZm),x,Sh,2,true))),x,4);

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

M00ftZm:=coeff(ser,x,0); M01Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,1)/t2Tm)); M02ftZm:=factor(subs(Tm=Zm^2,coeff(ser,x,2)/t2Tm));

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

LCQqMCIlQzUiIiIpSSNabUc2IiIiJEYlKSwmKiQpRiciIiNGJUYlRiVGJUYuRiUsJkYnRiVGJSEiIkYlLChGLEYlKiZGLkYlRidGJUYwRiVGMEYlKSwmRixGJUYpRiUiIidGMCwmRidGJUYlRiVGMEYw

LCQqLiIjWyIiIiksJkkjWm1HNiJGJUYlISIiIiIjRiUsLiomIiIkRiUpRigiIiZGJUYlKiYiI0JGJSlGKCIiJUYlRioqJiIjRUYlKUYoRi5GJUYlKiYiI0FGJSlGKEYrRiVGJSomRi5GJUYoRiVGJUYlRiVGJSksJiokRjpGJUYlRiVGJUYrRiUpLCZGPkYlRi5GJSIiJ0YqLCZGKEYlRiVGJUYqRiU=

Walks ending at (-1,0)

Cm10stsol:=factor(subs(Tm=Zm^2,M00ftZm/t2Tm));

LCQqKiIkYyMiIiIpSSNabUc2IiIiJ0YlLCgqJClGJyIiJUYlRiUqJkYpRiUpRiciIiNGJSEiIiIjNkYxRiUpLCYqJEYvRiVGJSIiJEYlRilGMUYx

n:=10: normal(series(subs(Zm=Zmser(n),Cm10stsol),t,n+1));

Ky9JInRHNiIiIiIiIiEiIigiIiMiI2wiIiUiJCNwIiInIiUiKnoiIiktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

coeff(coeff(convert(Cser(n),polynom),x,-1),y,0);

LCwqJiIlIip6IiIiKUkidEc2IiIiKkYlRiUqJiIkI3BGJSlGJyIiKEYlRiUqJiIjbEYlKUYnIiImRiVGJSomRi1GJSlGJyIiJEYlRiVGJ0Yl

Ending at (-1,1)

Cm11sol:=M01Zm;

LCQqMCIlQzUiIiIpSSNabUc2IiIiJEYlKSwmKiQpRiciIiNGJUYlRiVGJUYuRiUsJkYnRiVGJSEiIkYlLChGLEYlKiZGLkYlRidGJUYwRiVGMEYlKSwmRixGJUYpRiUiIidGMCwmRidGJUYlRiVGMEYw

n:=10: normal(series(subs(Zm=Zmser(n),Cm11sol),t,n+1));

Ky1JInRHNiIiIiNGJSIjQSIiJSIkXSMiIiciJSEqSCIiKS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzU=

coeff(coeff(convert(Cser(n),polynom),x,-1),y,1);

LCwqJiImZXMkIiIiKUkidEc2IiIjNUYlRiUqJiIlISpIRiUpRiciIilGJUYlKiYiJF0jRiUpRiciIidGJUYlKiYiI0FGJSlGJyIiJUYlRiUqJiIiI0YlKUYnRjdGJUYl

Ending at (-1,2)

Cm12ftsol:=M02ftZm:

n:=10: normal(series(subs(Zm=Zmser(n),Cm12ftsol),t,n+1));

Ky1JInRHNiIiIiQiIiUiI1siIiciJG0nIiIpIiVJISoiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

coeff(coeff(convert(Cser(n),polynom),x,-1),y,2);

LCoqJiIlSSEqIiIiKUkidEc2IiIiKkYlRiUqJiIkbSdGJSlGJyIiKEYlRiUqJiIjW0YlKUYnIiImRiVGJSomIiIkRiUpRidGM0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending on the negative x-axis: the series M_{i,0} (up to abscissa -4)</Text-field>

with(gfun):

ser:=map(factor,op(algeqtoseries(algRxtZm,x,Rxt,4,true)));

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

M00ftZm:=coeff(ser,x,0); M10Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,1)/t2Tm)); M20ftZm:=factor(subs(Tm=Zm^2,coeff(ser,x,2)/t2Tm)); M30Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,3)/t2Tm^2));

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

LCQqLCMiJGMjIiIkIiIiKUkjWm1HNiJGJkYnLDQqJClGKSIiKUYnRicqJkYmRicpRikiIihGJyEiIiomIiInRicpRilGNEYnRicqJiIjPUYnKUYpIiImRidGMiomIiIqRicpRikiIiVGJ0YnKiYiI0JGJ0YoRidGMiomRj1GJylGKSIiI0YnRicqJkY9RidGKUYnRjJGPUYyRicpLCYqJEZBRidGJ0YmRidGNEYyLCZGKUYnRidGJ0YyRjI=

LCQqLCMiIiIiIiRGJSwmSSNabUc2IkYlRiUhIiJGJSw8KiQpRigiIzdGJUYlKiYiIiNGJSlGKCIjNkYlRiUqJiIkMyJGJSlGKCIjNUYlRioqJiIkJUhGJSlGKCIiKkYlRiUqJiIkQihGJSlGKCIiKUYlRioqJiIlSzhGJSlGKCIiKEYlRiUqJiIkSyVGJSlGKCIiJ0YlRioqJiIlKz5GJSlGKCIiJkYlRiUqJiIkKkhGJSlGKCIiJUYlRiUqJiIkeSNGJSlGKEYmRiVGKiomIiRTJkYlKUYoRjBGJUYlKiYiJFkpRiVGKEYlRiUiJEIlRiVGJSksJiokRlRGJUYlRiZGJUZGRiosJkYoRiVGJUYlRipGJQ==

LCQqLiMiJUM1IiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiZGJkYnKUYpIiM4RidGJyomIiNCRicpRikiIzdGJ0YsKiYiI2dGJylGKSIjNkYnRicqJiIkSyJGJylGKSIjNUYnRiwqJiIkbiJGJylGKSIiKkYnRicqJiIjKipGJylGKSIiKUYnRiwqJiIkayNGJylGKSIiKEYnRicqJiIkLyJGJylGKSIiJ0YnRicqJiIjKClGJylGKSIiJkYnRiwqJiIjQEYnKUYpIiIlRidGLComRjZGJ0YoRidGJyomIiRjIkYnKUYpIiIjRidGJyomIiNYRidGKUYnRiciIzpGJ0YnKSwmKiRGWEYnRidGJkYnRkBGLCksJkYpRidGJ0YnRllGLEYn

Walks ending at (-1,0)

Cm10stsol:=factor(subs(Tm=Zm^2,M00ftZm/t2Tm));

LCQqKiIkYyMiIiIpSSNabUc2IiIiJ0YlLCgqJClGJyIiJUYlRiUqJkYpRiUpRiciIiNGJSEiIiIjNkYxRiUpLCYqJEYvRiVGJSIiJEYlRilGMUYx

n:=10: normal(series(subs(Zm=Zmser(n),Cm10stsol),t,n+1));

Ky9JInRHNiIiIiIiIiEiIigiIiMiI2wiIiUiJCNwIiInIiUiKnoiIiktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

coeff(coeff(convert(Cser(n),polynom),x,-1),y,0);

LCwqJiIlIip6IiIiKUkidEc2IiIiKkYlRiUqJiIkI3BGJSlGJyIiKEYlRiUqJiIjbEYlKUYnIiImRiVGJSomRi1GJSlGJyIiJEYlRiVGJ0Yl

Ending at (-2,0)

Cm20sol:=M10Zm-Q00/3;

LCYqLCMiJGMjIiIkIiIiKUkjWm1HNiJGJkYnLDQqJClGKSIiKUYnRicqJkYmRicpRikiIihGJyEiIiomIiInRicpRilGNEYnRicqJiIjPUYnKUYpIiImRidGMiomIiIqRicpRikiIiVGJ0YnKiYiI0JGJ0YoRidGMiomRj1GJylGKSIiI0YnRicqJkY9RidGKUYnRjJGPUYyRicpLCYqJEZBRidGJ0YmRidGNEYyLCZGKUYnRidGJ0YyRjIqJiNGJ0YmRidJJFEwMEdGKkYnRjI=

n:=12: normal(series(subs(Zm=Zmser(n),Q00=coeff(coeff(convert(Qser(n),polynom),x,0),y,0),Cm20sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIzYiIiUiJEMiIiInIiVyOSIiKSImKyM9IiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

coeff(coeff(convert(Cser(n),polynom),x,-2),y,0);

LC4qJiInInBLIyIiIilJInRHNiIiIzdGJUYlKiYiJisjPUYlKUYnIiM1RiVGJSomIiVyOUYlKUYnIiIpRiVGJSomIiRDIkYlKUYnIiInRiVGJSomIiM2RiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

Ending at (-3,0)

Cm30ftsol:=M20ftZm-t/3*Q10:

n:=12: normal(series(subs(Zm=Zmser(n),Q10=coeff(coeff(convert(Qser(n),polynom),x,1),y,0),Cm30ftsol),t,n+1));

Ky1JInRHNiIiIiIiIiUiIzsiIiciJD8jIiIpIiVgSCIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

coeff(coeff(convert(Cser(n),polynom),x,-3),y,0);

LCwqJiImXChSIiIiKUkidEc2IiIjNkYlRiUqJiIlYEhGJSlGJyIiKkYlRiUqJiIkPyNGJSlGJyIiKEYlRiUqJiIjO0YlKUYnIiImRiVGJSokKUYnIiIkRiVGJQ==

Ending at (-4,0)

Cm40sol:=M30Zm-Q20/3;

LCYqLiMiJUM1IiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiZGJkYnKUYpIiM4RidGJyomIiNCRicpRikiIzdGJ0YsKiYiI2dGJylGKSIjNkYnRicqJiIkSyJGJylGKSIjNUYnRiwqJiIkbiJGJylGKSIiKkYnRicqJiIjKipGJylGKSIiKUYnRiwqJiIkayNGJylGKSIiKEYnRicqJiIkLyJGJylGKSIiJ0YnRicqJiIjKClGJylGKSIiJkYnRiwqJiIjQEYnKUYpIiIlRidGLComRjZGJ0YoRidGJyomIiRjIkYnKUYpIiIjRidGJyomIiNYRidGKUYnRiciIzpGJ0YnKSwmKiRGWEYnRidGJkYnRkBGLCksJkYpRidGJ0YnRllGLEYnKiYjRidGJkYnSSRRMjBHRipGJ0Ys

n:=12: normal(series(subs(Zm=Zmser(n),Q20=coeff(coeff(convert(Qser(n),polynom),x,2),y,0),Cm40sol),t,n+1));

Ky1JInRHNiIiIiIiIiUiI0EiIiciJG0kIiIpIiV6YiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

coeff(coeff(convert(Cser(n),polynom),x,-4),y,0);

LCwqJiImXUEpIiIiKUkidEc2IiIjN0YlRiUqJiIlemJGJSlGJyIjNUYlRiUqJiIkbSRGJSlGJyIiKUYlRiUqJiIjQUYlKUYnIiInRiVGJSokKUYnIiIlRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0) </Text-field>

limit(limit(PexprM,x=0),y=0);

LCQqJiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiU2I0kiTUdGLTYkIiIhRjJGJUYl

P00Zm:=2*M10Zm;

LCQqLCMiJDcmIiIkIiIiKUkjWm1HNiJGJkYnLDQqJClGKSIiKUYnRicqJkYmRicpRikiIihGJyEiIiomIiInRicpRilGNEYnRicqJiIjPUYnKUYpIiImRidGMiomIiIqRicpRikiIiVGJ0YnKiYiI0JGJ0YoRidGMiomRj1GJylGKSIiI0YnRicqJkY9RidGKUYnRjJGPUYyRicpLCYqJEZBRidGJ0YmRidGNEYyLCZGKUYnRidGJ0YyRjI=

C00sol:=P00Zm+1/3*Q00;

LCYqLCMiJDcmIiIkIiIiKUkjWm1HNiJGJkYnLDQqJClGKSIiKUYnRicqJkYmRicpRikiIihGJyEiIiomIiInRicpRilGNEYnRicqJiIjPUYnKUYpIiImRidGMiomIiIqRicpRikiIiVGJ0YnKiYiI0JGJ0YoRidGMiomRj1GJylGKSIiI0YnRicqJkY9RidGKUYnRjJGPUYyRicpLCYqJEZBRidGJ0YmRidGNEYyLCZGKUYnRidGJ0YyRjIqJiNGJ0YmRidJJFEwMEdGKkYnRic=

n:=10: normal(series(subs(Zm=Zmser(n),Q00=coeff(coeff(convert(Qser(n),polynom),x,0),y,0),C00sol),t,n+1));

Ky9JInRHNiIiIiIiIiEiIiUiIiMiI0tGJyIkPSQiIiciJUlOIiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

coeff(coeff(convert(Cser(n),polynom),x,0),y,0);

LC4qJiImVz4lIiIiKUkidEc2IiIjNUYlRiUqJiIlSU5GJSlGJyIiKUYlRiUqJiIkPSRGJSlGJyIiJ0YlRiUqJiIjS0YlKUYnIiIlRiVGJSomRjVGJSlGJyIiI0YlRiVGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,1): the series M11</Text-field>

convert(series(convert(series(eqcatM,y,3),polynom),x,3),polynom);

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

coeff(coeff(%,x,1),y,0);

LCwqKCIiIyIiIkkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiRGJUYkNiNJIk1HRic2JCIiIUYzRiUhIiIqJkYmRiUtLSZGKzYkRiVGJUYwRjJGJUY0KiZGJEYlLS0mRis2I0YlRjBGMkYlRiUqKEYkRiVGJkYlLUYxRjJGJUY0I0YkIiIkRjQ=

expand(isolate(%,(D[1, 2](M))(0, 0)));

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkIiIiIiIjNiNJIk1HRio2JCIiIUYxLCoqJiNGLEYtRiwtLSZGJjYkRixGLEYuRjBGLCEiIiomSSJ0R0YqRjktLSZGJjYjRixGLkYwRixGLC1GL0YwRjkqJiNGLCIiJEYsRjtGOUY5

M11ftZm:=factor(subs(Tm=Zm^2,-M20ftZm-M00ftZm+M10Zm-1/3));

LCQqLCMiIiIiIiRGJSwmSSNabUc2IkYlRiUhIiJGJSw8KiQpRigiIzdGJUYlKiYiIiNGJSlGKCIjNkYlRiUqJiIkYyJGJSlGKCIjNUYlRioqJiIkKT5GJSlGKCIiKkYlRiUqJiIkeiZGJSlGKCIiKUYlRioqJiIlOzxGJSlGKCIiKEYlRiUqJiIjW0YlKUYoIiInRiVGJSomIiV3Q0YlKUYoIiImRiVGJSomIiVqNkYlKUYoIiIlRiVGJSomIiR1KUYlKUYoRiZGJUYlKiYiJDMiRiUpRihGMEYlRiUqJiIlcTZGJUYoRiVGKiIkJmVGKkYlKSwmKiRGVEYlRiVGJkYlRkZGKiwmRihGJUYlRiVGKkYl

Cm21ftsol:=M11ftZm-t/3*Q01;

LCYqLCMiIiIiIiRGJSwmSSNabUc2IkYlRiUhIiJGJSw8KiQpRigiIzdGJUYlKiYiIiNGJSlGKCIjNkYlRiUqJiIkYyJGJSlGKCIjNUYlRioqJiIkKT5GJSlGKCIiKkYlRiUqJiIkeiZGJSlGKCIiKUYlRioqJiIlOzxGJSlGKCIiKEYlRiUqJiIjW0YlKUYoIiInRiVGJSomIiV3Q0YlKUYoIiImRiVGJSomIiVqNkYlKUYoIiIlRiVGJSomIiR1KUYlKUYoRiZGJUYlKiYiJDMiRiUpRihGMEYlRiUqJiIlcTZGJUYoRiVGKiIkJmVGKkYlKSwmKiRGVEYlRiVGJkYlRkZGKiwmRihGJUYlRiVGKkYlKihGJEYlSSJ0R0YpRiVJJFEwMUdGKUYlRio=

n:=16: series(subs(Zm=Zmser(n+1),Q01=coeff(coeff(convert(Qser(n),polynom),x,0),y,1),Cm21ftsol),t,n+1);

KzNJInRHNiIiIiQiIiUiI1YiIiciJGYmIiIpIiVjcyIjNSImU2IqIiM3IigpZXk3IiM5IilOOlA8IiM7LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjPA==

coeff(coeff(convert(Cser(n),polynom),x,-2),y,1);

LDAqJiIpTjpQPCIiIilJInRHNiIiIzpGJUYlKiYiKClleTdGJSlGJyIjOEYlRiUqJiImU2IqRiUpRiciIzZGJUYlKiYiJWNzRiUpRiciIipGJUYlKiYiJGYmRiUpRiciIihGJUYlKiYiI1ZGJSlGJyIiJkYlRiUqJiIiJEYlKUYnRj9GJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,1)</Text-field>

series(convert(series(PexprM,x,2),polynom),y,3);;

KylJInlHNiIsJComIiIjIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKDYjSSJNR0YkNiQiIiFGNEYoRihGNCwmLS0mRiw2JEYoRidGMUYzRigqJiNGKEYnRigtLSZGLDYkRihGKEYxRjNGKEYoRigtSSJPR0YuRjBGJw==

P01ftZm:=factor(M20ftZm+M11ftZm);

LCQqLCMiIiMiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsPCokKUYpIiM3RidGJyomRiVGJylGKSIjNkYnRicqJiIkSyJGJylGKSIjNUYnRisqJiIkWSNGJylGKSIiKkYnRicqJiIkXidGJylGKSIiKUYnRisqJiIlQzpGJylGKSIiKEYnRicqJiIkIz5GJylGKSIiJ0YnRisqJiIlKT0jRicpRikiIiZGJ0YnKiYiJEooRicpRikiIiVGJ0YnKiYiJClIRicpRilGJkYnRicqJiIkQyRGJylGKUYlRidGJyomIiRpIkYnRilGJ0YrIiMiKUYrRicpLCYqJEZURidGJ0YmRidGRkYrLCZGKUYnRidGJ0YrRic=

C01ftsol:=P01ftZm+t*Q01/3;

LCYqLCMiIiMiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsPCokKUYpIiM3RidGJyomRiVGJylGKSIjNkYnRicqJiIkSyJGJylGKSIjNUYnRisqJiIkWSNGJylGKSIiKkYnRicqJiIkXidGJylGKSIiKUYnRisqJiIlQzpGJylGKSIiKEYnRicqJiIkIz5GJylGKSIiJ0YnRisqJiIlKT0jRicpRikiIiZGJ0YnKiYiJEooRicpRikiIiVGJ0YnKiYiJClIRicpRilGJkYnRicqJiIkQyRGJylGKUYlRidGJyomIiRpIkYnRilGJ0YrIiMiKUYrRicpLCYqJEZURidGJ0YmRidGRkYrLCZGKUYnRidGJ0YrRicqKCNGJ0YmRidJInRHRipGJ0kkUTAxR0YqRidGJw==

n:=16: series(subs(Zm=Zmser(n+1),Q01=coeff(coeff(convert(Qser(n),polynom),x,0),y,1),C01ftsol),t,n+1);

KzVJInRHNiIiIiIiIiMiIioiIiUiIyUqIiInIiV0NSIiKSImIilIIiIjNSInLk87IiM3IihzYjcjIiM5IilrP0ZHIiM7LUkiT0clKnByb3RlY3RlZEc2I0YlIiM8

coeff(coeff(convert(Cser(n),polynom),x,0),y,1);

LDIqJiIpaz9GRyIiIilJInRHNiIiIzpGJUYlKiYiKHNiNyNGJSlGJyIjOEYlRiUqJiInLk87RiUpRiciIzZGJUYlKiYiJiIpSCJGJSlGJyIiKkYlRiUqJiIldDVGJSlGJyIiKEYlRiUqJiIjJSpGJSlGJyIiJkYlRiUqJkY1RiUpRiciIiRGJUYlRidGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,1): the series M_{2,1}</Text-field>

series(convert(series(eqcatM,x,4),polynom),y,3);

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

coeff(coeff(convert(%,polynom),x,2),y,0);

LCoqKCMiIiIiIiRGJUkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiVGJUYlRiU2I0kiTUdGKDYkIiIhRjRGJSEiIiomRidGJS0tJkYsNiVGJUYlIiIjRjFGM0YlRjUtLSZGLDYkRiVGJUYxRjNGJSooRjtGJUYnRiUtLSZGLDYjRiVGMUYzRiVGNQ==

expand(isolate(%,(D[1, 1, 2](M))(0, 0)));

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlIiIiRiwiIiM2I0kiTUdGKjYkIiIhRjEsKComI0YsIiIkRiwtLSZGJjYlRixGLEYsRi5GMEYsISIiKiZJInRHRipGOi0tJkYmNiRGLEYsRi5GMEYsRiwqJkYtRiwtLSZGJjYjRixGLkYwRixGOg==

M21Zm:=factor(subs(Tm=Zm^2,-2*M30Zm+2*M20ftZm/t2Tm-2*M10Zm)/2);

LCQqLiMiJDcmIiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiQpRikiIzhGJ0YnKiYiIidGJylGKSIjN0YnRiwqJiIiJkYnKUYpIiM2RidGJyomIiNTRicpRikiIzVGJ0YsKiYiI11GJylGKSIiKkYnRicqJiIjZ0YnKUYpIiIpRidGLComIiRxJEYnKUYpIiIoRidGJyomIiMpKUYnKUYpRjJGJ0YsKiYiJHAnRicpRilGNkYnRicqJiIkKT5GJylGKSIiJUYnRiwqJiIkJFJGJ0YoRidGJyomIiRXIkYnKUYpIiIjRidGLComIiNbRidGKUYnRiciI0NGJ0YnLCZGKUYnRidGJ0YsKSwmKiRGV0YnRidGJkYnRkBGLEYn

Cm31sol:=M21Zm-Q11/3;

LCYqLiMiJDcmIiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiQpRikiIzhGJ0YnKiYiIidGJylGKSIjN0YnRiwqJiIiJkYnKUYpIiM2RidGJyomIiNTRicpRikiIzVGJ0YsKiYiI11GJylGKSIiKkYnRicqJiIjZ0YnKUYpIiIpRidGLComIiRxJEYnKUYpIiIoRidGJyomIiMpKUYnKUYpRjJGJ0YsKiYiJHAnRicpRilGNkYnRicqJiIkKT5GJylGKSIiJUYnRiwqJiIkJFJGJ0YoRidGJyomIiRXIkYnKUYpIiIjRidGLComIiNbRidGKUYnRiciI0NGJ0YnLCZGKUYnRidGJ0YsKSwmKiRGV0YnRidGJkYnRkBGLEYnKiYjRidGJkYnSSRRMTFHRipGJ0Ys

n:=14: series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),Cm31sol),t,n+1);

Ky9JInRHNiIiIiVGJSIjdSIiJyIlOzYiIikiJnFmIiIjNSInM2BBIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,-3),y,1);

LC4qJiIoa1c8JCIiIilJInRHNiIiIzlGJUYlKiYiJzNgQUYlKUYnIiM3RiVGJSomIiZxZiJGJSlGJyIjNUYlRiUqJiIlOzZGJSlGJyIiKUYlRiUqJiIjdUYlKUYnIiInRiVGJSomIiIlRiUpRidGO0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">And now walks ending at (1,1)</Text-field>

series(convert(series(PexprM,x,3),polynom),y,3);;

KylJInlHNiIsJiomIiIjIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKDYjSSJNR0YkNiQiIiFGNEYoRigqJiwmLS0mRiw2JEYoRidGMUYzRigqJiNGKEYnRigtLSZGLDYkRihGKEYxRjNGKEYoRihJInhHRiRGKEYoRjQsKEY3RihGO0YoKiYtLSZGLDYlRihGKEYnRjFGM0YoRkFGKEYoRigtSSJPR0YuRjBGJw==

coeff(coeff(convert(%,polynom),x,1),y,1);

LS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUiIiJGKyIiIzYjSSJNR0YpNiQiIiFGMA==

P11Zm:=2*M21Zm;

LCQqLiMiJUM1IiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiQpRikiIzhGJ0YnKiYiIidGJylGKSIjN0YnRiwqJiIiJkYnKUYpIiM2RidGJyomIiNTRicpRikiIzVGJ0YsKiYiI11GJylGKSIiKkYnRicqJiIjZ0YnKUYpIiIpRidGLComIiRxJEYnKUYpIiIoRidGJyomIiMpKUYnKUYpRjJGJ0YsKiYiJHAnRicpRilGNkYnRicqJiIkKT5GJylGKSIiJUYnRiwqJiIkJFJGJ0YoRidGJyomIiRXIkYnKUYpIiIjRidGLComIiNbRidGKUYnRiciI0NGJ0YnLCZGKUYnRidGJ0YsKSwmKiRGV0YnRidGJkYnRkBGLEYn

C11sol:=P11Zm+Q11/3;

LCYqLiMiJUM1IiIkIiIiKUkjWm1HNiJGJkYnLCZGKUYnRichIiJGJyw+KiQpRikiIzhGJ0YnKiYiIidGJylGKSIjN0YnRiwqJiIiJkYnKUYpIiM2RidGJyomIiNTRicpRikiIzVGJ0YsKiYiI11GJylGKSIiKkYnRicqJiIjZ0YnKUYpIiIpRidGLComIiRxJEYnKUYpIiIoRidGJyomIiMpKUYnKUYpRjJGJ0YsKiYiJHAnRicpRilGNkYnRicqJiIkKT5GJylGKSIiJUYnRiwqJiIkJFJGJ0YoRidGJyomIiRXIkYnKUYpIiIjRidGLComIiNbRidGKUYnRiciI0NGJ0YnLCZGKUYnRidGJ0YsKSwmKiRGV0YnRidGJkYnRkBGLEYnKiYjRidGJkYnSSRRMTFHRipGJ0Yn

n:=14: series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),C11sol),t,n+1);

KzFJInRHNiIiIiNGJSIjQyIiJSIkKUciIiciJXdOIiIpIiYrZSUiIzUiJ0M7ZyIjNy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzk=

coeff(coeff(convert(Cser(n),polynom),x,1),y,1);

LDAqJiIoV20xKSIiIilJInRHNiIiIzlGJUYlKiYiJ0M7Z0YlKUYnIiM3RiVGJSomIiYrZSVGJSlGJyIjNUYlRiUqJiIld05GJSlGJyIiKUYlRiUqJiIkKUdGJSlGJyIiJ0YlRiUqJiIjQ0YlKUYnIiIlRiVGJSomIiIjRiUpRidGP0YlRiU=

<Text-field style="Heading 2" layout="Heading 2">Looking for nice numbers</Text-field>

For some mysterious reasons (a bug with the command hypergeomsols) I have to restart the session here

restart:

algTm:=256*Tm^3*t^2-Tm^4-8*Tm^3-18*Tm^2+27;

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

algZm:=subs(Tm=Zm^2,algTm);

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(algTm,t,Tm,n,true)): end:

Tmser(5);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

<Text-field style="Heading 3" layout="Heading 3">What about walks ending at (-1,0) ? no </Text-field>

Cm10stsol:=-256*Zm^6*(Zm^4-6*Zm^2-11)/(Zm^2+3)^6;

LCQqKiIkYyMiIiIpSSNabUc2IiIiJ0YlLCgqJClGJyIiJUYlRiUqJkYpRiUpRiciIiNGJSEiIiIjNkYxRiUpLCYqJEYvRiVGJSIiJEYlRilGMUYx

n:=5: series(subs(Zm=Zmser(n),Cm10stsol),t,n+1);

KytJInRHNiIiIiIiIiEiIigiIiMiI2wiIiUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIiY=

algCm10st:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm10st-Cm10stsol),algZm,Zm))));

LCQqJiJHL1YkWzsjWyEpPXZYImZ4JVFZOFlwSSIiIiksPCooIiYkbz5GJSlJJ0NtMTBzdEc2IiIiJUYlKUkic0dGLCIiJ0YlRiUqKCImU1AlRiUpRisiIiRGJSlGLyIiJkYlRiUqKCIlKD0jRiVGM0YlKUYvRi1GJSEiIiooIiZlUSRGJSlGKyIiI0YlRjlGJUYlKigiJXBSRiVGPUYlKUYvRjRGJUY6KigiIyIpRiVGPUYlKUYvRj5GJUYlKigiJks2IkYlRitGJUZBRiVGJSooIiVYQEYlRitGJUZERiVGOiooIiQ5IkYlRitGJUYvRiVGJSomIiVKOEYlRkRGJUYlRitGOiomIiQyIkYlRi9GJUY6RiVGJUY+RiVGJQ==

algeqtodiffeq(op(1,op(2,algCm10st)),Cm10st(s));

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

de:=%:

rec:=diffeqtorec(de,Cm10st(s),co(k));

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

rec1:=map(factor,op(1,%));

LCwqLCImT2InIiIiLCYqJiIiJEYlSSJrRzYiRiVGJSIiJkYlRiUsJkYnRiUiIiVGJUYlLCZGKUYlRiVGJUYlLUkjY29HRio2I0YpRiVGJSooIiVbP0YlLCgqJiIjYUYlKUYpIiIjRiVGJSomIiRLI0YlRilGJUYlIiRWI0YlRiUtRjA2I0YuRiVGJSoqIiRvKEYlLCYqJkY4RiVGKUYlRiUiIihGJUYlLCgqJkYoRiVGN0YlRiUqJiIjQEYlRilGJUYlIiNKRiVGJS1GMDYjLCZGKUYlRjhGJUYlISIiKigiIilGJSwoRjVGJSomIiRDJkYlRilGJUYlIiVsN0YlRiUtRjA2IywmRilGJUYoRiVGJUZLKiosJkYnRiUiIzxGJUYlLCZGJ0YlIiM7RiVGJSwmRilGJSIiJ0YlRiUtRjA2IywmRilGJUYtRiVGJUYl

No HG solution

with(LREtools):hypergeomsols(rec1,co(k),{co(0) = 1, co(1) = 7, co(2) = 65, co(3) = 692});

hypergeomsols(rec1,co(k),{});

IiIh

In fact there seems to be a shorter recurrence relation

n:=70: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm10stsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

PCUsKComLC4qJiIlOyMqIiIiKUkia0c2IiIiJkYoRigqJiImZ0AqRigpRioiIiVGKEYoKiYiJ2dYTkYoKUYqIiIkRihGKComIidTLWxGKClGKiIiI0YoRigqJiInayFvJkYoRipGKEYoIidTJSo9RihGKC1JI2NvR0YrNiNGKkYoRigqJiwuKiYiJV82RihGKUYoISIiKiYiJitXIkYoRi9GKEZDKiYiJiEpMyhGKEYzRihGQyomIicrOzxGKEY3RihGQyomIiczWT9GKEYqRihGQyImP2wqRkNGKC1GPTYjLCZGKkYoRihGKEYoRigqJiwuKiYiI09GKEYpRihGKComIiRTJkYoRi9GKEYoKiYiJSY9JEYoRjNGKEYoKiYiJU4jKkYoRjdGKEYoKiYiJiU+OEYoRipGKEYoIiUhWyhGKEYoLUY9NiMsJkYqRihGOEYoRihGKC8tRj02IyIiIUYoLy1GPTYjRigiIig=

rec1:=map(factor,op(1,rec));

LCgqLiIkYyMiIiIsJiomIiIkRiVJImtHNiJGJUYlIiImRiVGJSwmRidGJSIiJUYlRiUsJkYpRiVGJUYlRiUsKComRi1GJSlGKSIiI0YlRiUqJiIjQ0YlRilGJUYlIiNQRiVGJS1JI2NvR0YqNiNGKUYlRiUqKiIiKUYlLCYqJkYyRiVGKUYlRiVGK0YlRiUsLComIiNzRiUpRilGLUYlRiUqJiIkPyhGJSlGKUYoRiVGJSomIiVJRUYlRjFGJUYlKiYiJV1URiVGKUYlRiUiJThDRiVGJS1GNzYjRi5GJSEiIiosLCZGJ0YlIiM1RiVGJSwmRilGJUYtRiVGJSwmRidGJSIjNkYlRiUsKEYwRiUqJiIjO0YlRilGJUYlIiM8RiVGJS1GNzYjLCZGKUYlRjJGJUYlRiU=

hypergeomsols(rec1,co(k),{co(0) = 1, co(1) = 7});

hypergeomsols(rec1,co(k),{});

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,1) : no</Text-field>

Cm11sol := -1024*Zm^3*(Zm^2+1)^2*(Zm-1)*(Zm^2-2*Zm-1)/((Zm^2+3)^6*(Zm+1));

LCQqMCIlQzUiIiIpSSNabUc2IiIiJEYlKSwmKiQpRiciIiNGJUYlRiVGJUYuRiUsJkYnRiVGJSEiIkYlLChGLEYlKiZGLkYlRidGJUYwRiVGMEYlKSwmRixGJUYpRiUiIidGMCwmRidGJUYlRiVGMEYw

n:=10: series(subs(Zm=Zmser(n),Cm11sol),t,n+1);

Ky1JInRHNiIiIiNGJSIjQSIiJSIkXSMiIiciJSEqSCIiKS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzU=

algCm11:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm11-Cm11sol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm11,Cm11(s)):nops(de0);

IiIj

op(2,de0);

Ly1JJUNtMTFHNiI2IyIiIUYn

de:=de0:

rec:=diffeqtorec(de,Cm11(s),co(k)):

rec1:=map(factor,op(1,%));

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

{seq(op(i,rec),i=2..nops(rec))};

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIiIjLy1GJTYjRi0iI0EvLUYlNiMiIiQiJF0jLy1GJTYjIiIlIiUhKkgvLUYlNiMiIiYiJmVzJC8tRiU2IyIiJyInTyN6JS8tRiU2IyIiKCIoayY+ai8tRiU2IyIiKSIpSTYsJikvLUYlNiMiIioiKy02S2k2Ly1GJTYjIiM1Iiw7Qip6NTsvLUYlNiMiIzYiLWVcQil5RCMvLUYlNiMiIzciLmlWKFIkZT4kLy1GJTYjIiM4Ii9XTGBCZGhYLy1GJTYjIiM5IjBxSSp5IUgpZWw=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

LCYqOiMiJEciIiV4OCIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRictRi02IyNGK0YpISIiLCYqJiIjWkYnLUkjY29HRjE2IyIiIUYnRicqJkY1RictRj42I0YnRidGOUYnKUkjUGlHRi9GKkY5KSIjO0kia0dGMUYnLUYtNiMsJkZIRicjIiIoRjVGJ0YnLUYtNiMsJkZIRidGKkYnRictRi02IywmRkhGJyNGTUYpRidGOS1GLTYjLCZGSEYnRilGJ0Y5LCgqJClGSEYrRidGJyomI0Y0RitGJ0ZIRidGJyMiI0giI0NGJ0YnRicqOCMiI1MiJGYlRidGNkYnLCYqJiIkXyVGJ0Y9RidGJyomIiMoKUYnRkJGJ0Y5RidGREY5RixGOUZGRidGTkYnLUYtNiMsJkZIRidGM0YnRidGVUY5LUYtNiMsJkZIRicjIiIpRilGJ0Y5LCgqJiMiIiUiIzpGJ0ZaRidGJyomRjhGJ0ZIRidGJyNGJyIiKkYnRidGOQ==

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm11sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 2, co(2) = 22, co(3) = 250}));

hypergeomsols(rec1,co(k),{});

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0) : hypergeometric</Text-field>

P00Zm:= -(512/3)*Zm^3*(Zm^8-3*Zm^7+6*Zm^6-18*Zm^5+9*Zm^4-23*Zm^3+4*Zm^2-4*Zm-4)/((Zm^2+3)^6*(Zm+1));

algP00:=subs(t=sqrt(s),factor(resultant(numer(P00-P00Zm),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algP00,P00(s)));

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

rec:=diffeqtorec(de,P00(s),co(k)):

rec1:=map(factor,op(1,rec));

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

{seq(op(i,rec),i=2..nops(rec))};

PCkvLUkjY29HNiI2IyIiISMiIiMiIiQvLUYlNiMiIiIjIiM1RisvLUYlNiNGKiMiIycpRisvLUYlNiNGKyMiJCUpKUYrLy1GJTYjIiIlIiVNTC8tRiU2IyIiJiImJzRTLy1GJTYjIiInIidNSl0=

HG solution

map(factor,hypergeomsols(rec1,co(k),%));

LCYqOCMiI0siJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJykiIztJImtHRjFGJy1GLTYjLCZGOEYnIyIiKEY1RidGJy1GLTYjLCZGOEYnRipGJ0YnLCgqJiIjQ0YnKUY4RitGJ0YnKiYiI2dGJ0Y4RidGJyIjSEYnRicpSSNQaUdGL0YqISIiLUYtNiMjRitGKUZKLUYtNiMsJkY4RicjRj1GKUYnRkotRi02IywmRjhGJ0YpRidGSkYnKjYjRjdGJkYnRktGJ0Y2RidGPkYnLUYtNiMsJkY4RidGM0YnRicsKComIiM3RidGREYnRicqJiIjSUYnRjhGJ0YnRjRGJ0YnRkhGSkYsRkpGUkZKLUYtNiMsJkY4RicjIiIpRilGJ0ZKRko=

In fact there seems to be a shorter recurrence relation

n:=70: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),P00Zm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

map(factor,hypergeomsols(rec1,co(k),{co(0) = 2/3, co(1) = 10/3}));

The ascending factorial

fact:=proc(a,n) if n=0 then 1 else a*fact(a+1,n-1): fi: end:

seq((32/729)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+7/6)*GAMMA(k+1/2)*(24*k^2+60*k+29)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+7/3)*GAMMA(k+3))/(16^k*(24*k^2+60*k+29)*fact(1/2,k)*fact(7/6,k)/fact(2,k+1)/fact(4/3,k+1)),k=1..10);

NiwjIiM7IiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

seq((16/729)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)*(12*k^2+30*k+5)/(sqrt(Pi)*GAMMA(5/6)*GAMMA(k+3)*GAMMA(k+8/3))/ ((12*k^2+30*k+5)*fact(1/2,k)*fact(5/6,k)*16^k/fact(2,k+1)/fact(5/3,k+1)),k=1..10);

NiwjIiIpIiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

paper expression

pap:=proc(n) 4*16^n/3^5*( 3^4*fact(1/2,n)*fact(1/2,n+1)/fact(2,n)/fact(2,n+1) + 4*(24*n^2+60*n+29)* fact(1/2,n)*fact(7/6,n)/fact(2,n+1)/fact(4/3,n+1) -2*(12*n^2+30*n+5)*fact(1/2,n)*fact(5/6,n)/fact(2,n+1)/fact(5/3,n+1)): end:

seq(pap(n),n=0..10);

Ni0iIiIiIiUiI0siJD0kIiVJTiImVz4lIic1P18iKD0hPm4iKVNvcykpIitBeV8mPiIiLDluWXRqIg==

coeff(coeff(convert(Cser(20),polynom),x,0),y,0);

LDgqJiIsOW5ZdGoiIiIiKUkidEc2IiIjP0YlRiUqJiIrQXlfJj4iRiUpRiciIz1GJUYlKiYiKVNvcykpRiUpRiciIztGJUYlKiYiKD0hPm5GJSlGJyIjOUYlRiUqJiInNT9fRiUpRiciIzdGJUYlKiYiJlc+JUYlKUYnIiM1RiVGJSomIiVJTkYlKUYnIiIpRiVGJSomIiQ9JEYlKUYnIiInRiVGJSomIiNLRiUpRiciIiVGJUYlKiZGSUYlKUYnIiIjRiVGJUYlRiU=

seq(4*16^n*fact(1/2,n)*fact(1/2,n+1)/fact(2,n)/fact(2,n+1)/Qconj(0,0,n),n=1..10);

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

Asymptotic behaviour

map(simplify,expand(asympt((32/729)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+7/6)*GAMMA(k+1/2)*(24*k^2+60*k+29)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+7/3)*GAMMA(k+3)),k,3)));

LCYqMCMiJEciIiMiKSIiIikiIztJImtHNiJGJykiIiMjRiciIiRGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YrNiMjRi1GL0YnSSNQaUdGMyEiIkYqRjgpKiRGKkY4RjZGJ0YnKiZGKEYnLUkiT0dGMzYjKiYpRipGLUY4RjlGJ0YnRic=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0) : Hypergeometric too</Text-field>

M10Zm := -(256/3)*Zm^3*(Zm^8-3*Zm^7+6*Zm^6-18*Zm^5+9*Zm^4-23*Zm^3+4*Zm^2-4*Zm-4)/((Zm^2+3)^6*(Zm+1));

n:=12: series(subs(Zm=Zmser(n),M10Zm),t,n+1);

KzFJInRHNiIjIiIiIiIkIiIhIyIiJkYnIiIjIyIjVkYnIiIlIyIkVSVGJyIiJyIlbjsiIikiJlsrIyIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjNw==

algM10:=factor(subs(t=sqrt(s),factor(resultant(numer(M10-M10Zm),algZm,Zm)))):

algeqtodiffeq(algM10,M10(s));

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

de:=%:

rec:=diffeqtorec(de,M10(s),co(k)):

rec1:=map(factor,op(1,%));

{seq(op(i,rec),i=2..nops(rec))};

PCkvLUkjY29HNiI2IyIiISMiIiIiIiQvLUYlNiNGKiMiIiZGKy8tRiU2IyIiIyMiI1ZGKy8tRiU2I0YrIyIkVSVGKy8tRiU2IyIiJSIlbjsvLUYlNiNGMCImWysjLy1GJTYjIiInIiduOkQ=

HG solution

map(factor,hypergeomsols(rec1,co(k),%));

LCYqOCMiIzsiJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJylGJUkia0dGMUYnLUYtNiMsJkY3RicjIiIoRjVGJ0YnLUYtNiMsJkY3RidGKkYnRicsKComIiNDRicpRjdGK0YnRicqJiIjZ0YnRjdGJ0YnIiNIRidGJylJI1BpR0YvRiohIiItRi02IyNGK0YpRkktRi02IywmRjdGJyNGPEYpRidGSS1GLTYjLCZGN0YnRilGJ0ZJRicqNiMiIilGJkYnRkpGJ0Y2RidGPUYnLUYtNiMsJkY3RidGM0YnRicsKComIiM3RidGQ0YnRicqJiIjSUYnRjdGJ0YnRjRGJ0YnRkdGSUYsRklGUUZJLUYtNiMsJkY3RicjRlZGKUYnRklGSQ==

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),M10Zm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

rec1:=map(factor,op(1,rec));

(hypergeomsols(rec1,co(k),{co(0) = 1/3, co(1) = 5/3}));

LCYqOCMiJEciIiRWIyIiIilJI1BpRyUqcHJvdGVjdGVkRyNGJyIiIyEiIikiIiRGK0YnLUkmR0FNTUFHNiRGKkkoX3N5c2xpYkc2IjYjI0YsRi9GLS1GMTYjIyIiJiIiJ0YnKSIjO0kia0dGNEYnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RidGK0YnRictRjE2IywmRj5GJyNGQ0YvRidGLS1GMTYjLCZGPkYnRi9GJ0YtLCgqJClGPkYsRidGJyomI0Y6RixGJ0Y+RidGJyMiI0giI0NGJ0YnRicqNiMiI1MiJEgoRidGKEYtRjdGLUYwRidGPEYnRkRGJy1GMTYjLCZGPkYnRjlGJ0YnRktGLS1GMTYjLCZGPkYnIyIiKUYvRidGLSwoKiYjIiM3RjpGJ0ZQRidGJyomRjtGJ0Y+RidGJ0YnRidGJ0Yt

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,0) : no</Text-field>

M20ftZm := (1/3)*(Zm-1)*(Zm^12+2*Zm^11-108*Zm^10+294*Zm^9-723*Zm^8+1332*Zm^7-432*Zm^6+1900*Zm^5+299*Zm^4-278*Zm^3+540*Zm^2+846*Zm+423)/((Zm^2+3)^6*(Zm+1));

n:=12: series(subs(Zm=Zmser(n),M20ftZm),t,n+1);

Ky9JInRHNiIjIiIiIiIkIiIjIyIiKUYnIiIlIyIjJClGJyIiJyIkPSRGKiIleFEiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIzc=

algM20ft:=factor(subs(t=sqrt(s),factor(resultant(numer(M20ft-M20ftZm),algZm,Zm)))):

de0:=algeqtodiffeq(algM20ft,M20ft(s)): nops(%);

IiIj

op(2,de0);

Ly1JJk0yMGZ0RzYiNiMiIiFGJw==

de:=de0: :

rec:=diffeqtorec(de,M20ft(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PDQvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIyIiKUYuLy1GJTYjRi4jIiMkKUYuLy1GJTYjIiIlIiQ9JC8tRiU2IyIiJiIleFEvLUYlNiMiIiciJig9XC8tRiU2IyIiKCInJVxVJy8tRiU2I0Y0IyIpRipbZCNGLi8tRiU2IyIiKiMiKipcIz1dJEYuLy1GJTYjIiM1IyIrZiZIRCRbRi4vLUYlNiMiIzYjIiwqR0BlXW5GLi8tRiU2IyIjNyMiLSJHVzB3XypGLi8tRiU2IyIjOCIuP2AzXUBfJS8tRiU2IyIjOSIvY2AsJT0oKVsnLy1GJTYjIiM6IjAmZitOPkZ0JCovLUYlNiMiIzsiMicpPkNgP3c/TyIvLUYlNiMiIzwiM0Zsa3BXQnkqKT4=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

map(factor,hypergeomsols(rec1,co(k),{}));

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

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),M20ftZm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k))):

rec1:=map(factor,op(1,rec)):

map(factor,hypergeomsols(rec1,co(k),{}));

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-4,0) : no</Text-field>

M30Zm := (1024/3)*Zm^3*(Zm-1)*(3*Zm^13-23*Zm^12+60*Zm^11-132*Zm^10+167*Zm^9-99*Zm^8+264*Zm^7+104*Zm^6-87*Zm^5-21*Zm^4+60*Zm^3+156*Zm^2+45*Zm+15)/((Zm^2+3)^9*(Zm+1)^2);

n:=12: series(subs(Zm=Zmser(n),M30Zm),t,n+1);

Ky9JInRHNiIjIiIiIiIkIiIjIiIlRikiI10iIiciJFknIiIpIiVcJikiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIzc=

algM30:=factor(subs(t=sqrt(s),factor(resultant(numer(M30-M30Zm),algZm,Zm)))):

de0:=algeqtodiffeq(algM30,M30(s)): nops(%);

IiNH

op(2,de0);

LCQqJiI9KyNmMU09JSk0Oy1MVkA6JyIiIilJInNHNiIiIz5GJSEiIg==

de:=de0: :

rec:=diffeqtorec(de,M30(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

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

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

map(factor,hypergeomsols(rec1,co(k),{}));

LCYqOiMiI1MiKVY+WjwiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLCYqJiIkVidGJy1JI2NvR0YxNiNGJ0YnRicqJiIlbEVGJy1GOjYjIiIhRichIiJGJykiIztJImtHRjFGJy1GLTYjLCZGREYnIyIiKEY1RidGJy1GLTYjLCZGREYnRipGJ0YnLCwqJiIkJ3BGJylGRCIiJUYnRicqJiIlS1ZGJylGREYpRidGJyomIiVIIipGJylGREYrRidGJyomIiVvd0YnRkRGJ0YnIiUiKT5GJ0YnLUYtNiMjRitGKUZBKUkjUGlHRi9GKkZBLUYtNiMsJkZERicjIiM1RilGJ0ZBLUYtNiMsJkZERidGUUYnRkFGJyo4RiRGJ0ZlbkYnLCYqJiIlOjlGJ0Y5RidGJyomIiYuPiJGJ0Y+RidGQUYnRkJGJ0ZKRictRi02IywmRkRGJ0YzRidGJywsKiYiJFskRidGUEYnRicqJiIlZT9GJ0ZURidGJyomIiUyVEYnRldGJ0YnKiYiJUZSRidGREYnRiciJSdHIkYnRidGaG5GQUYsRkFGX29GQS1GLTYjLCZGREYnIyIjNkYpRidGQUZB

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,2) : no</Text-field>

M02ftZm := 48*(Zm-1)^2*(3*Zm^5-23*Zm^4+26*Zm^3+22*Zm^2+3*Zm+1)*(Zm^2+1)^2/((Zm^2+3)^6*(Zm+1));

Cm12ftsol := %:

n:=12: series(subs(Zm=Zmser(n),Cm12ftsol),t,n+1);

Ky9JInRHNiIiIiQiIiUiI1siIiciJG0nIiIpIiVJISoiIzUiJyJwQSIiIzctSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

algCm12ft:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm12ft-Cm12ftsol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm12ft,Cm12ft(s)): nops(%);

IiIj

op(2,de0);

Ly1JJ0NtMTJmdEc2IjYjIiIhRic=

de:=de0: :

rec:=diffeqtorec(de,Cm12ft(s),co(k)):

rec1:=collect(op(1,rec),co,factor):

nops(rec);

IiM+

{seq(op(i,rec),i=2..nops(rec))};

PDQvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMiIiQvLUYlNiNGMSIjWy8tRiU2IyIiJSIkbScvLUYlNiMiIiYiJUkhKi8tRiU2IyIiJyInInBBIi8tRiU2IyIiKCIobjlvIi8tRiU2IyIiKSIpdmhGQi8tRiU2IyIiKiIqZGxPRCQvLUYlNiMiIzUiK21QRiplJS8tRiU2IyIjNiIsM3JBZ18nLy1GJTYjIiM3Ii1QKFw2IlskKi8tRiU2IyIjOCIveVoucSV5TSIvLUYlNiMiIzkiMFNEWydII1smPi8tRiU2IyIjOiIxSTtDJDRoLCZHLy1GJTYjIiM7IjIlPmRmJXpkYTwlLy1GJTYjIiM8IjMkKilbITQ0RF9WaA==

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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

In fact there seems to be a shorter recurrence relation

n:=220: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm12ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k))):

rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0)=0,co(1)=0,co(2)=3,co(3)=48,co(4)=666}));

hypergeomsols(rec1,co(k),{});

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

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,1) : no</Text-field>

P01ftZm := (2/3)*(Zm-1)*(Zm^12+2*Zm^11-132*Zm^10+246*Zm^9-651*Zm^8+1524*Zm^7-192*Zm^6+2188*Zm^5+731*Zm^4+298*Zm^3+324*Zm^2-162*Zm-81)/((Zm^2+3)^6*(Zm+1));

algP01ft:=subs(t=sqrt(s),factor(resultant(numer(P01ft-P01ftZm),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algP01ft,P01ft(s))):

nops(de);

IiIj

op(2,de);

Ly1JJlAwMWZ0RzYiNiMiIiFGJw==

rec:=diffeqtorec(de,P01ft(s),co(k)):

rec1:=map(factor,op(1,rec)):

{seq(op(i,rec),i=2..nops(rec))};

PDQvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiIyIiJC8tRiU2I0YuIyIjQUYvLy1GJTYjRi8jIiRaI0YvLy1GJTYjIiIlIiR2Ki8tRiU2IyIiJiImZD8iLy1GJTYjIiInIidsVDovLUYlNiMiIigiKEZMLSMvLUYlNiMiIikjIilpKVI4KUYvLy1GJTYjIiIqIyIreGFlMzZGLy8tRiU2IyIjNSMiLG0qcEBLOkYvLy1GJTYjIiM2IyItIjNoSkc5I0YvLy1GJTYjIiM3IyIuLzU4QXEtJEYvLy1GJTYjIiM4Ii8nPWg4UXhWIi8tRiU2IyIjOSIwQEFoLUlUMSMvLUYlNiMiIzoiMVRzcmhANSQpSC8tRiU2IyIjOyIyKT08KkdAUWxMJS8tRiU2IyIjPCIzZSh5JkhXOi5Qag==

Some HG solutions but not ours

(hypergeomsols(rec1,co(k),%));

(hypergeomsols(rec1,co(k),{}));

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

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),P01ftZm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k))):

rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 2/3, co(2) = 22/3, co(3) = 247/3, co(4) = 975}));

(hypergeomsols(rec1,co(k),{}));

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,1) : HG solution (also, consequently, for walks ending at (-3,1))</Text-field>

P11Zm := (1024/3)*Zm^3*(Zm-1)*(Zm^13-6*Zm^12+5*Zm^11-40*Zm^10+50*Zm^9-60*Zm^8+370*Zm^7-88*Zm^6+669*Zm^5-198*Zm^4+393*Zm^3-144*Zm^2+48*Zm+24)/((Zm+1)*(Zm^2+3)^9);

algP11:=subs(t=sqrt(s),factor(resultant(numer(P11-P11Zm),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algP11,P11(s)));

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

rec:=diffeqtorec(de,P11(s),co(k)):

rec1:=map(factor,op(1,rec));

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

{seq(op(i,rec),i=2..nops(rec))};

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJSIiJC8tRiU2IyIiIyMiI2NGLy8tRiU2I0YvIyIkQyhGLy8tRiU2I0YuIiVHSi8tRiU2IyIiJiImIT1ULy1GJTYjIiInIicpR14mLy1GJTYjIiIoIihzU1woLy1GJTYjIiIpIyIqJSlwdTQkRi8vLUYlNiMiIioiK3MwPFI5

HG solution

map(factor,hypergeomsols(rec1,co(k),%));

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

In fact there seems to be a shorter recurrence relation

n:=120: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),P11Zm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

map(factor,hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 4/3}));

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,1) : no</Text-field>

M11ftZm := (1/3)*(Zm-1)*(Zm^12+2*Zm^11-156*Zm^10+198*Zm^9-579*Zm^8+1716*Zm^7+48*Zm^6+2476*Zm^5+1163*Zm^4+874*Zm^3+108*Zm^2-1170*Zm-585)/((Zm^2+3)^6*(Zm+1));

n:=12: series(subs(Zm=Zmser(n),M11ftZm),t,n+1);

Ky9JInRHNiIjIiIiIiIkIiIjIyIjOUYnIiIlIyIkayJGJyIiJyIkZCciIikiJSE9KSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjNw==

algM11ft:=factor(subs(t=sqrt(s),factor(resultant(numer(M11ft-M11ftZm),algZm,Zm)))):

de0:=algeqtodiffeq(algM11ft,M11ft(s)): nops(%);

IiIj

op(2,de0);

Ly1JJk0xMWZ0RzYiNiMiIiFGJw==

de:=de0 :

rec:=diffeqtorec(de,M11ft(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PDQvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIyIjOUYuLy1GJTYjRi4jIiRrIkYuLy1GJTYjIiIlIiRkJy8tRiU2IyIiJiIlIT0pLy1GJTYjIiInIid5XDUvLUYlNiMiIigiKEwzUSIvLUYlNiMiIikjIilONGZiRi4vLUYlNiMiIiojIip5SFNlKEYuLy1GJTYjIiM1IyIsMi9rKls1Ri4vLUYlNiMiIzYjIi0jelJ0eFkiRi4vLUYlNiMiIzcjIi5CbmVoVTIjRi4vLUYlNiMiIzgiLm1lRkpfJikqLy1GJTYjRjQiMGxvZj1lX1QiLy1GJTYjIiM6IjFZbUBvXHhYPy8tRiU2IyIjOyIyLV9uditpVyhILy1GJTYjIiM8IjNKQSQqZio+XHNNJQ==

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),M11ftZm),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k))):

rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1/3, co(2) = 14/3, co(3) = 164/3}));

hypergeomsols(rec1,co(k),{});

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<Text-field style="Heading 2" layout="Heading 2">The total number of walks</Text-field>

subs(x=1,y=1,eqC);

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We need to expand R(1) at t=1/4

degree(algRTm,Rx);

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res:=factor(subs(x=1,algRTm)):nops(res);

IiIl

op(3,res);

LCZJI1RtRzYiIiIiRiUhIiI=

nops(op(4,res));

IiRBJA==

degree(op(4,res),Rx);

IiIn

res1:=factor(resultant(subs(Rx=R1,op(4,res)),algTm,Tm)): nops(res1);

IiIm

seq(nops(op(i,res1)),i=1..nops(res1)); seq(indets(op(i,res1)),i=1..nops(res1));

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n:=10: series(subs(R1=subs(x=1,Rser(n)),op(4,res1)),t,n+1);

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series(subs(R1=subs(x=1,Rser(n)),op(5,res1)),t,n+1);

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algR1:=collect(op(1,op(5,res1)),R1,factor);

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with(algcurves):

genus(algR1,t,R1);

IiIi

The possible singularities

coeff(algR1,R1,8);

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factor(discrim(algR1,R1));

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{fsolve(%)};

PCckISsrKysrRCEjNSQiIiFGJyQiK0s8cHlzISM2JCIrKysrK0RGJSQiK0d6cSdlKUYl

with(gfun):

algeqtoseries(subs(t=(1-epsilon)/4,algR1),epsilon,R1,2,true);

NyQsKCMiIiMiIiQiIiIqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCgqJiIiKkYnKUkjX1pHRitGJUYnRicqJiIjQ0YnRjRGJyEiIiIjS0YnRidJKGVwc2lsb25HRi5GJ0YnLUkiT0dGLDYjKiQpRjkjRjIiIilGJ0YnLChGJEYnKiYtRio2IywmKiYiKTIqW1YiRicpRjQiIidGJ0YnIik7c3g7RjdGJylGOSNGJ0YmRidGJy1GOzYjKiQpRjkjRiZGQEYnRic=

solve(14348907*_Z^6-16777216,_Z);

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with(plots):

We have indeed singularities at 1/4 and -1/4

implicitplot(algR1,t=-1/4-0.1..1/4+0.1,R1=-0.1..1,numpoints=10000);

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The singularity at t=-1/4 does not contribute, at first order

algeqtoseries(subs(t=-(1-epsilon)/4,algR1),epsilon,R1,2,true);

NyQrKUkoZXBzaWxvbkc2IiEiIyIiISMhIiUiIiQiIiItSSJPRyUqcHJvdGVjdGVkRzYjRisiIiMsKC1JJ1Jvb3RPZkc2JEYuSShfc3lzbGliR0YlNiMsKComIiRWI0YrKUkjX1pHRjRGMEYrRisqJiIkcypGK0Y7RitGKyIjXyEiIkYrKiYtRjM2IywoKiZGOUYrKUY7RipGK0YrKiYiIztGK0YyRitGKyIjS0YrRispRiQjRjBGKkYrRistRi02I0YkRis=

allvalues(RootOf(243*_Z^2+972*_Z-52));

NiQsJiIiIyEiIiomIyIjSyIjRiIiIikiIiQjRipGJEYqRiosJkYkRiVGJkYl

allvalues(RootOf(243*_Z^3+16*RootOf(243*_Z^2+972*_Z-52)+32));

NigsJiomIyIiJSIjRiIiIikiIiQjRigiIiNGKEYoKiYjRiYiIipGKF4jRihGKEYoLCQqJiMiIilGJ0YoRilGKCEiIiwmRiRGKEYtRjUsJEYyRigsJkYkRjVGLUYoLCZGJEY1Ri1GNQ==

n:=100:ser:=subs(x=1,y=1,Cser(n)):

plot([seq([i,evalf(coeff(ser,t,i)/4^i*i^(1/3))],i=1..n)],style=point);

NiotJSdDVVJWRVNHNiQ3YHE3JCQiIzUhIiIkIiM1ISIiNyQkIiM/ISIiJCIrPjRWLTYhIio3JCQiI0khIiIkIit2ISkqb0AiISIqNyQkIiNTISIiJCIrc3E6UzchIio3JCQiI10hIiIkIitaNiVlSCIhIio3JCQiI2chIiIkIislKjNHLzghIio3JCQiI3EhIiIkIis2XHNROCEiKjckJCIjISkhIiIkIisnM2pATSIhIio3JCQiIyEqISIiJCIrQys6bTghIio3JCQiJCsiISIiJCIrQmdXbjghIio3JCQiJDUiISIiJCIqQChRJlEiISIpNyQkIiQ/IiEiIiQiKyVIU2NRIiEiKjckJCIkSSIhIiIkIip4PCgqUiIhIik3JCQiJFMiISIiJCIrZVVVKlIiISIqNyQkIiRdIiEiIiQiK2BrJjNUIiEiKjckJCIkZyIhIiIkIitSX0U1OSEiKjckJCIkcSIhIiIkIitaRno+OSEiKjckJCIkIT0hIiIkIit0IVEhPjkhIio3JCQiJCE+ISIiJCIrei05RjkhIio3JCQiJCsjISIiJCIrMSUpSEU5ISIqNyQkIiQ1IyEiIiQiKykzLExWIiEiKjckJCIkPyMhIiIkIitmcFRLOSEiKjckJCIkSSMhIiIkIitrLWJROSEiKjckJCIkUyMhIiIkIitFNWxQOSEiKjckJCIkXSMhIiIkIit2RjNWOSEiKjckJCIkZyMhIiIkIitjXT1VOSEiKjckJCIkcSMhIiIkIisoKTMvWjkhIio3JCQiJCFHISIiJCIrJypbOlk5ISIqNyQkIiQhSCEiIiQiKyopNGBdOSEiKjckJCIkKyQhIiIkIitDSG1cOSEiKjckJCIkNSQhIiIkIitIVmpgOSEiKjckJCIkPyQhIiIkIiokeXlfOSEiKTckJCIkSSQhIiIkIisnMzlrWCIhIio3JCQiJFMkISIiJCIrVzZmYjkhIio3JCQiJF0kISIiJCIrcywjKmU5ISIqNyQkIiRnJCEiIiQiKzA8N2U5ISIqNyQkIiRxJCEiIiQiK3REPmg5ISIqNyQkIiQhUSEiIiQiKyp6PS9ZIiEiKjckJCIkIVIhIiIkIipzaktZIiEiKTckJCIkKyUhIiIkIisuV15pOSEiKjckJCIkNSUhIiIkIis2LTtsOSEiKjckJCIkPyUhIiIkIitEW1ZrOSEiKjckJCIkSSUhIiIkIit2UyFwWSIhIio3JCQiJFMlISIiJCIrOD4/bTkhIio3JCQiJF0lISIiJCIrRlBebzkhIio3JCQiJGclISIiJCIrdFIkeVkiISIqNyQkIiRxJSEiIiQiKm8vK1oiISIpNyQkIiQhWyEiIiQiK2xrTXA5ISIqNyQkIiQhXCEiIiQiKzEsUnI5ISIqNyQkIiQrJiEiIiQiK1BEdnE5ISIqNyQkIiQ1JiEiIiQiK2s3b3M5ISIqNyQkIiQ/JiEiIiQiK1tNMXM5ISIqNyQkIiRJJiEiIiQiK095KVFaIiEiKjckJCIkUyYhIiIkIisoKikpR3Q5ISIqNyQkIiRdJiEiIiQiKy4jPV1aIiEiKjckJCIkZyYhIiIkIisoR1BXWiIhIio3JCQiJHEmISIiJCIrXCd6Z1oiISIqNyQkIiQhZSEiIiQiK1FmXnY5ISIqNyQkIiQhZiEiIiQiK1omeXFaIiEiKjckJCIkKychIiIkIitgN2B3OSEiKjckJCIkNSchIiIkIipcPyF5OSEiKTckJCIkPychIiIkIitsKSlbeDkhIio3JCQiJEknISIiJCIrOC8iKnk5ISIqNyQkIiRTJyEiIiQiK1hQUnk5ISIqNyQkIiRdJyEiIiQiKydvXyh6OSEiKjckJCIkZychIiIkIisqSF0jejkhIio3JCQiJHEnISIiJCIrJT5eMFsiISIqNyQkIiQhbyEiIiQiK1xDMSFbIiEiKjckJCIkIXAhIiIkIisuJTQ4WyIhIio3JCQiJCsoISIiJCIrJnBMM1siISIqNyQkIiQ1KCEiIiQiKzovLiNbIiEiKjckJCIkPyghIiIkIit0cmMiWyIhIio3JCQiJEkoISIiJCIrPHFyI1siISIqNyQkIiRTKCEiIiQiKypwbEFbIiEiKjckJCIkXSghIiIkIis9PFAkWyIhIio3JCQiJGcoISIiJCIrOT0kSFsiISIqNyQkIiRxKCEiIiQiKyZ5JypSWyIhIio3JCQiJCF5ISIiJCIrN3ljJFsiISIqNyQkIiQheiEiIiQiKkYlZiVbIiEiKTckJCIkKykhIiIkIit1ZDwlWyIhIio3JCQiJDUpISIiJCIrUWc7JlsiISIqNyQkIiQ/KSEiIiQiKydlZFpbIiEiKjckJCIkSSkhIiIkIisleThkWyIhIio3JCQiJFMpISIiJCIqKFxKJlsiISIpNyQkIiRdKSEiIiQiK2QhUmlbIiEiKjckJCIkZykhIiIkIislXFxlWyIhIio3JCQiJHEpISIiJCIrckt1J1siISIqNyQkIiQhKSkhIiIkIismZmlqWyIhIio3JCQiJCEqKSEiIiQiK0J4QShbIiEiKjckJCIkKyohIiIkIispZWJvWyIhIio3JCQiJDUqISIiJCIrLE9wKFsiISIqNyQkIiQ/KiEiIiQiKm9IdFsiISIpNyQkIiRJKiEiIiQiKy0/OSlbIiEiKjckJCIkUyohIiIkIislKWZ5KFsiISIqNyQkIiRdKiEiIiQiK0tSZClbIiEiKjckJCIkZyohIiIkIitDYkEpWyIhIio3JCQiJHEqISIiJCIrQy4qKilbIiEiKjckJCIkISkqISIiJCIrWSNcJylbIiEiKjckJCIkISoqISIiJCIrUD9SKlsiISIqNyQkIiUrNSEiIiQiK0MhZSEqWyIhIiotJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0mJSZfQVhJU0c2IyIiIjYnLSUrX0dSSURMSU5FU0c2Jy0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lKV9WSVNJQkxFRzYjIiIjLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNictJStfR1JJRExJTkVTRzYnLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUpX1ZJU0lCTEVHNiMiIictJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkUSE2IlEhNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRJJSEiIi0lKUJPVU5EU19ZRzYjJCIjcSEiIi0lLUJPVU5EU19XSURUSEc2IyQiJT9NISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJSFcJCEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJSFSJCEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU/TiEiIi0lKUNISUxEUkVORzYiRzYi

evalf(2^5/3^3*sqrt(3)/GAMMA(2/3));

JCIrLiJwZl4iISIq

In passing, we check the asymptotic behaviour for C00

ser:=coeff(coeff(convert(Cser(n),polynom),x,0),y,0);

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

plot([seq([i,evalf(coeff(ser,t,i)/4^i*i^(5/3))],i=3..n)],style=point);

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

evalf(2^9/3^4*GAMMA(2/3)/Pi);

JCIrIXpIWHMjISIq

<Text-field style="Heading 1" layout="Heading 1">Starting at (-1,0)</Text-field>

Functional equation. The series Ch(1/x) is negative in x and Cv(1/y) is negative in y

K:=1-t*(x+1/x+y+1/y):

eqC:=-K*C(x,y)+1/x-t/y*Ch(1/x)-t/x*Cv(1/y);

LCoqJiwmIiIiRiUqJkkidEc2IkYlLCpJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiVGJUYsRiUtSSJDR0YoNiRGKkYtRiVGLEYrRiUqKEYnRiVGLUYsLUkjQ2hHRig2I0YrRiVGLCooRidGJUYqRiwtSSNDdkdGKDYjRi5GJUYs

Chser:=proc(n) local res:res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end:

Cvser:=proc(n) local res:res:=coeff(convert(Cser(n),polynom),x,0): add(coeff(res,y,-i)/y^i,i=1..n+2): end:

Cser:=proc(n) option remember: if n=0 then 1/x else normal(series(subs(C(x,y)=Cser(n-1),Ch(1/x)=Chser(n-1),Cv(1/y)=Cvser(n-1),eqC+C(x,y)),t,n+1)): fi: end:

map(expand,Cser(2));coeff(map(expand,Cser(7)),t,7);

KylJInRHNiIqJEkieEdGJCEiIiIiISwoIiIiRioqJkYmRidJInlHRiRGKkYqKiQpRiYiIiNGJ0YqRiosMEYmRioqJkYvRipGLEYqRioqJkYmRicpRixGL0YqRioqJEYsRidGKiomIiIkRipGJkYnRioqKEYvRipGLkYnRixGKkYqKiQpRiZGNkYnRipGLw==

LFxxIiQjcCIiIiooIiQiZUYkSSJ4RzYiRiRJInlHRihGJEYkKigiJEkpRiRGJyEiIkYpRiRGJCooIiIoRiRGKUYkKUYnIiImRiRGJCooIiRJI0YkRidGJClGKSIiJEYkRiQqKCIjQEYkRidGJClGKUYwRiRGJCooIiNORiQpRidGNEYkRjNGJEYkKihGOUYkKUYnIiIjRiQpRikiIiVGJEYkKigiJEQjRiRGPEYkKUYpRj1GJEYkKigiJCdSRiRGJ0YsRjNGJEYkKigiJCFRRiRGJ0YkRilGLEYkKigiJCVmRiRGPEYsRkJGJEYkKigiJCVcRiRGOkYsRilGJEYkKigiI1tGJEYnRixGN0YkRiQqKCIkMSJGJEY6RiRGKUYsRiQqKCIkUyJGJEY8RixGPkYkRiQqKCIkTCJGJEY8RiRGQkYsRiQqKCIkQyNGJEY6RixGM0YkRiQqKCIjKClGJEYnRiRGM0YsRiQqKCIkNSNGJClGJ0Y/RixGQkYkRiQqKCIkNyJGJEYvRixGKUYkRiQqJkYnRiwpRilGLkYkRiQqKCIiJ0YkRi9GJEYpRixGJCooRi5GJEY8RiwpRilGaW5GJEYkKigiIzpGJEZZRiRGQkYsRiQqKEY2RiRGOkYsRjdGJEYkKigiIz9GJEY6RiRGM0YsRiQqKEY5RiRGWUYsRj5GJEYkKihGXW9GJEY8RiRGPkYsRiQqKEY5RiRGL0YsRjNGJEYkKihGaW5GJEYnRiRGN0YsRiQqKEY2RiQpRidGaW5GLEZCRiRGJCokRltvRixGJCooRi5GJClGJ0YuRixGKUYkRiQqJClGJyIiKUYsRiQqKEY2RiRGWUYkRkJGJEYkKiYiJEkkRiRGPEYkRiQqJiIkPidGJEZCRiRGJComIiQ3JkYkRjxGLEYkKiYiI1ZGJEZZRiRGJComIiRUIkYkRj5GJEYkKiYiJCg+RiRGWUYsRiQqJiIkIT1GJEZCRixGJCokRmZvRiRGJComRi5GJEZbb0YkRiQqJiIjR0YkRmZvRixGJComIiNERiRGPkYsRiQqKCIkSyJGJEY6RiRGKUYkRiQ=

The orbit sum vanishes

n:=6: normal(series(x*y*Cser(n)-y/x*subs(x=1/x,Cser(n))-x/y*subs(y=1/y,Cser(n))+1/x/y*subs(x=1/x,y=1/y,Cser(n)),t,n+1));

IiIh

L denotes the NW part of C (L for "left").

B stands for "below" and corresponds to the SE part of C

Lser:=proc(n) collect(add(1/x*coeff(convert(Cser(n),polynom),x,-i)*x^i,i=1..n+1),t,normal): end:

Lser(3);

LCoiIiJGIyomLC4qJClJInhHNiIiIiRGI0YjKihGKkYjKUYoIiIjRiNJInlHRilGI0YjKihGKkYjRihGIylGLkYtRiNGIyokKUYuRipGI0YjKiYiIidGI0YoRiNGIyomIiIpRiNGLkYjRiNGIylJInRHRilGKkYjRiMqJiwqKiRGLEYjRiMqKEYtRiNGKEYjRi5GI0YjKiRGMEYjRiNGKkYjRiMpRjhGLUYjRiMqJiwmRi5GI0YoRiNGI0Y4RiNGIw==

Bser:=proc(n) collect(add(1/y*coeff(convert(Cser(n),polynom),y,-i)*y^i,i=1..n),t,normal): end:

Bser(6);

LCwqJiw0KiYiIiYiIiIpSSJ4RzYiIiIlRidGJyooIiM1RicpRikiIiRGJ0kieUdGKkYnRicqKEYtRicpRikiIiNGJylGMEYzRidGJyooRiZGJ0YpRicpRjBGL0YnRicqJClGMEYrRidGJyomIiNnRidGMkYnRicqKCIjYEYnRilGJ0YwRidGJyomIiM+RidGNEYnRiciJDMiRidGJylJInRHRioiIidGJ0YnKiYsLiomRitGJ0YuRidGJyooRkJGJ0YyRidGMEYnRicqKEYrRidGKUYnRjRGJ0YnKiRGNkYnRicqJiIjSEYnRilGJ0YnKiYiIzlGJ0YwRidGJ0YnKUZBRiZGJ0YnKiYsKiomRi9GJ0YyRidGJyooRi9GJ0YpRidGMEYnRicqJEY0RidGJ0YtRidGJylGQUYrRidGJyomLCYqJkYzRidGKUYnRidGMEYnRicpRkFGL0YnRicqJClGQUYzRidGJw==

P is the NE part of C (P as positive)

Pser:=proc(n) local res: res:=convert(Cser(n),polynom): collect(add(add(coeff(coeff(res,x,i),y,j)*x^i*y^j,i=0..n),j=0..n),t): end:

Pser(3);

LCgqJiwqKiQpSSJ4RzYiIiIjIiIiRioqKCIiJEYqRidGKkkieUdGKEYqRioqJkYsRiopRi1GKUYqRioiIihGKkYqKUkidEdGKEYsRipGKiomLCZGJ0YqKiZGKUYqRi1GKkYqRiopRjJGKUYqRipGMkYq

<Text-field style="Heading 2" layout="Heading 2">Quadrant-like equations for L and B</Text-field>

eqorbitC:=x*y*C(x,y)-y/x*C(1/x,y)-x/y*C(x,1/y)+1/x/y*C(1/x,1/y);

LCoqKEkieEc2IiIiIkkieUdGJUYmLUkiQ0dGJTYkRiRGJ0YmRiYqKEYnRiZGJCEiIi1GKTYkKiRGJEYsRidGJkYsKihGJEYmRidGLC1GKTYkRiQqJEYnRixGJkYsKihGJEYsRidGLC1GKTYkRi9GM0YmRiY=

We plug in the orbit equation the expression of C

Cexpr:=P(x,y)+1/x*L(1/x,y)+1/y*B(x,1/y);

LCgtSSJQRzYiNiRJInhHRiVJInlHRiUiIiIqJkYnISIiLUkiTEdGJTYkKiRGJ0YrRihGKUYpKiZGKEYrLUkiQkdGJTYkRicqJEYoRitGKUYp

eqo:=expand(subs(C(x,y)=Cexpr,C(1/x,y)=subs(x=1/x,Cexpr), C(x,1/y)=subs(y=1/y,Cexpr),C(1/x,1/y)=subs(x=1/x,y=1/y,Cexpr),eqorbitC));

LDoqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYtSSJMR0YlNiQqJEYkISIiRidGJkYmKiZGJEYmLUkiQkdGJTYkRiQqJEYnRjBGJkYmKihGJEYwRidGJi1GKUYuRiZGMComRidGJi1GLUYqRiZGMComRiRGMC1GMzYkRi9GNUYmRjAqKEYkRiZGJ0YwLUYpRjRGJkYwKiZGJ0YwLUYtRjxGJkYwKiZGJEYmLUYzRipGJkYwKihGJ0YwRiRGMC1GKUY8RiZGJiomRidGMC1GLUY0RiZGJiomRiRGMC1GM0YuRiZGJg==

normal(eqo+subs(x=1/x,eqo));

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Positive part in x and y

eqospxy:=x*y*P(x, y)-y*L(x, y)+y*L(0,y)-x*B(x, y)+x*B(x,0);

LCwqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYtSSJMR0YlRipGJiEiIiomRidGJi1GLTYkIiIhRidGJkYmKiZGJEYmLUkiQkdGJUYqRiZGLiomRiRGJi1GNTYkRiRGMkYmRiY=

This implies that the OS vanishes

Express P in terms of L and B

Pexpr:= expand(solve(eqospxy,P(x,y)));

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Checking the equivalence with the OS

CexprLB:=subs(P(x,y)=Pexpr,Cexpr);

LC4qJkkieUc2IiEiIi1JIkJHRiU2JEkieEdGJSIiISIiIkYmKiZGKkYmLUkiTEdGJTYkRipGJEYsRiwqJkYkRiYtRihGMEYsRiwqJkYqRiYtRi82JEYrRiRGLEYmKiZGKkYmLUYvNiQqJEYqRiZGJEYsRiwqJkYkRiYtRig2JEYqKiRGJEYmRixGLA==

normal(subs(C(x,y)=CexprLB,C(1/x,y)=subs(x=1/x,CexprLB), C(x,1/y)=subs(y=1/y,CexprLB), C(1/x,1/y)=subs(x=1/x,y=1/y, CexprLB),eqorbitC));

IiIh

We go back to the equation defining C

eqLB:=collect(subs(C(x,y)=CexprLB, Ch(1/x)=1/x*L(1/x,0), Cv(1/y)=1/y*B(0,1/y),eqC),[B,L]);

LDQqKkkidEc2IiIiIkkieEdGJSEiIkkieUdGJUYoLUkiQkdGJTYkIiIhKiRGKUYoRiZGKCooLCZGJkYmKiZGJEYmLCpGJ0YmKiRGJ0YoRiZGKUYmRi5GJkYmRihGJkYpRigtRis2JEYnRi1GJkYmKihGMEYmRilGKC1GKzYkRidGKUYmRigqKEYwRiZGKUYoLUYrNiRGJ0YuRiZGKCooRjBGJkYnRigtSSJMR0YlNiRGLUYpRiZGJiooRjBGJkYnRigtRj5GOEYmRigqKEYwRiZGJ0YoLUY+NiRGM0YpRiZGKEYzRiYqKkYkRiZGKUYoRidGKC1GPjYkRjNGLUYmRig=

Checking

indets(eqLB);

PC1JInRHNiJJInhHRiRJInlHRiQtSSJCR0YkNiQiIiEqJEYmISIiLUYoNiRGJUYqLUYoNiRGJUYmLUYoNiRGJUYrLUkiTEdGJDYkRipGJi1GNEYwLUY0NiQqJEYlRixGKi1GNDYkRjlGJg==

n:=10: normal(series(subs( B(x,1/y)=subs(y=1/y,Bser(n)), B(x,0)=subs(y=0,Bser(n)), B(0,1/y)=subs(x=0,y=1/y,Bser(n)), L(0,y)=subs(x=0,Lser(n)), L(1/x,y)=subs(x=1/x,Lser(n)), L(1/x,0)=subs(x=1/x,y=0,Lser(n)), L(x,y)=Lser(n),B(x,y)=Bser(n), eqLB),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

There are four bivariate series: B(x,y), B(x,1/y), L(x,y) and L(1/x,y).

We extract the negative part in x

eqLB;

Contribution of the series in x

factor(series(subs(L(1/x,y)=0,L(1/x,0)=0,eqLB),x,2));

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The series in 1/x

op(7,eqLB)+op(9,eqLB);

LCYqKCwmIiIiRiUqJkkidEc2IkYlLCpJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiVGJUYsRiVGKkYsLUkiTEdGKDYkRitGLUYlRiwqKkYnRiVGLUYsRipGLC1GMDYkRisiIiFGJUYs

factor(series(subs(x=1/x,%),x,2));

KylJInhHNiIqJkkidEdGJCIiIi1JIkxHRiQ2JCIiIUkieUdGJEYnRissJComLCwqKEYoRidGJkYnKUYsIiIjRichIiIqKEYmRictLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnNiNGKUYqRidGLEYnRjMqJkYmRictRik2JEYrRitGJ0YnRiVGMyomRixGJ0YoRidGJ0YnRixGM0YzRictSSJPR0Y6RjxGMg==

eqLBxneg:=collect(convert(%%%,polynom)+op(7,eqLB)+op(9,eqLB)-coeff(%,x,0),[L,B],factor);

LDAqJkkidEc2IiIiIi1JIkxHRiU2JCIiIUkieUdGJUYmISIiKipGJEYmRitGLEkieEdGJUYsLUYoNiQqJEYuRixGKkYmRiwqKiwsKihGJEYmKUYuIiIjRiZGK0YmRiYqKEYkRiZGLkYmKUYrRjZGJkYmKiZGJEYmRi5GJkYmKiZGJEYmRitGJkYmKiZGLkYmRitGJkYsRiYtRig2JEYxRitGJkY1RixGK0YsRiYqKEYuRixGJEYmLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJjYjRihGKUYmRiYqKkYkRiZGK0YsRi5GLC1JIkJHRiU2JEYqRipGJkYsKipGJEYmRitGLEYuRiwtRkpGKUYmRiZGMUYm

Checking that what remains is non-negative in x

normal(series(normal(eqLB-eqLBxneg),x,3));

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

... and that the negative part is indeed x-negative

normal(series(subs(x=1/x,eqLBxneg),x,3));

KylJInhHNiIsJComLDIqKC1JIkxHRiQ2JCIiIUkieUdGJCIiIkkidEdGJEYuKUYtIiIjRi4hIiIqKkYxRi5GL0YuLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLjYjRipGK0YuRi1GLkYyKiZGL0YuLUYqNiRGLEYsRi5GLiomRi9GLi1JIkJHRiRGP0YuRi4qJkYvRi4tRkJGK0YuRjIqJkYvRi5GKUYuRjIqJkYtRi5GKUYuRi5GLUYyRi5GLUYyRjJGLiwkKigjRi5GMUYuLC4qKkYxRi5GL0YuRjRGLkYwRi5GMiooLS0mRjc2JEYuRi5GPEYrRi5GL0YuRi1GLkYyKipGMUYuRi9GLkYpRi5GLUYuRjIqKEYxRi5GL0YuLUY1Rj9GLkYuKihGMUYuRi9GLkY0Ri5GMiooRjFGLkY0Ri5GLUYuRi5GLkYtRjJGMkYxLUkiT0dGOUY7IiIk

This gives us an expression of 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

D1Lsol:=collect(solve(coeff(%,x),(D[1](L))(0, y)),[L,B],factor);

LCwqKCMiIiIiIiNGJUkieUc2IiEiIi1JIkxHRig2JCIiIUYtRiVGJSosRiRGJSwoKiZJInRHRihGJSlGJ0YmRiVGJUYxRiVGJ0YpRiVGJ0YpRjFGKS1GKzYkRi1GJ0YlRikqKEYkRiVGJ0YpLUkiQkdGKEYsRiVGJSooRiRGJUYnRiktRjdGNEYlRikqJkYkRiVGMUYpRik=

eqLBxneg:=collect(subs((D[1](L))(0, y)=D1Lsol,eqLBxneg),[L,B,L0],factor);

LDAqLCMiIiIiIiNGJUkidEc2IkYlSSJ5R0YoISIiSSJ4R0YoRiotSSJMR0YoNiQiIiFGL0YlRiUqLEYkRiUsKioqRiZGJUYnRiVGK0YlRilGJUYlKiZGJ0YlKUYpRiZGJUYlRidGJUYpRipGJUYrRipGKUYqLUYtNiRGL0YpRiVGKioqRidGJUYpRipGK0YqLUYtNiQqJEYrRipGL0YlRioqKiwsKihGJ0YlKUYrRiZGJUYpRiVGJSooRidGJUYrRiVGNEYlRiUqJkYnRiVGK0YlRiUqJkYnRiVGKUYlRiUqJkYrRiVGKUYlRipGJS1GLTYkRjpGKUYlRj5GKkYpRipGJSosRiRGJUYnRiVGKUYqRitGKi1JIkJHRihGLkYlRioqLEYkRiVGJ0YlRilGKkYrRiotRkdGNkYlRiUqJkYkRiVGK0YqRiU=

paper version

eqpap:=1/x*Ker*(L(1/x,y)-L(0,y)/2)-1/2/x+t/x/y*L(1/x,0)-t*(1/x^2-1)*L(0,y)/2-t/x/y*B(0,y)/2-t/x/y*L(0,0)/2+t/x/y*B(0,0)/2;

LDAqKEkieEc2IiEiIkkkS2VyR0YlIiIiLCYtSSJMR0YlNiQqJEYkRiZJInlHRiVGKComI0YoIiIjRigtRis2JCIiIUYuRihGJkYoRigqJkYwRihGJEYmRiYqKkkidEdGJUYoRi5GJkYkRiYtRis2JEYtRjRGKEYoKipGMEYoRjdGKCwmKiQpRiRGMUYmRihGKEYmRihGMkYoRiYqLEYwRihGN0YoRi5GJkYkRiYtSSJCR0YlRjNGKEYmKixGMEYoRjdGKEYuRiZGJEYmLUYrNiRGNEY0RihGJiosRjBGKEY3RihGLkYmRiRGJi1GQEZDRihGKA==

factor(subs(Ker=K,eqLBxneg+eqpap));

IiIh

eqL:=collect(1/x*subs(x=1/x,-eqpap*2),[Ker,L,B]);

LDAqJiwmKiYiIiMiIiItSSJMRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKihJInRHRipGJ0YtRi4tRik2JEYxRjFGJ0YnKipGLEYuRjRGJywmKiQpRixGJkYnRidGJ0YuRidGL0YnRicqKkYmRidGNEYnRi1GLi1GKTYkRixGMUYnRi4qKEY0RidGLUYuLUkiQkdGKkY2RidGLkYnRicqKEY0RidGLUYuLUZARjBGJ0Yn

n:=10: normal(series(subs( Ker=K,L(0,0)=subs(x=0,y=0,Lser(n)), B(0,y)=subs(x=0,Bser(n)),B(0,0)=subs(x=0,y=0,Bser(n)), B(0,1/y)=subs(x=0,y=1/y,Bser(n)), L(0,y)=subs(x=0,Lser(n)), L(x,0)=subs(y=0,Lser(n)), L(x,y)=Lser(n), eqL),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

We get back to eqLB, fand extract the negative part in y

map(factor,eqLB);

LDQqKkkidEc2IiIiIkkieEdGJSEiIkkieUdGJUYoLUkiQkdGJTYkIiIhKiRGKUYoRiZGKCoqLCwqKEYkRiYpRiciIiNGJkYpRiZGJiooRiRGJkYnRiYpRilGM0YmRiYqJkYkRiZGJ0YmRiYqJkYkRiZGKUYmRiYqJkYnRiZGKUYmRihGJi1GKzYkRidGLUYmRidGKEY1RihGKCoqRjBGJi1GKzYkRidGKUYmRidGKEY1RihGJioqRjBGJi1GKzYkRidGLkYmRidGKEY1RihGJioqRjBGJi1JIkxHRiU2JEYtRilGJkYyRihGKUYoRigqKkYwRiYtRkNGPUYmRjJGKEYpRihGJioqRjBGJi1GQzYkKiRGJ0YoRilGJkYyRihGKUYoRiZGSkYmKipGJEYmRilGKEYnRigtRkM2JEZKRi1GJkYo

The series in y

normal(series(subs(B(x, 1/y)=0,B(0, 1/y)=0,eqLB),y,2));

KydJInlHNiIqKEkidEdGJCIiIiwoKiYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIzYjSSJCR0YkNiRJInhHRiQiIiFGJ0Y2RidGJy1JIkxHRiQ2JEY3RjchIiItRjlGNUYnRidGNkY7RjstSSJPR0YvNiNGJ0Y3

The series in 1/y

op(1,eqLB)+op(4,eqLB);

LCYqKkkidEc2IiIiIkkieEdGJSEiIkkieUdGJUYoLUkiQkdGJTYkIiIhKiRGKUYoRiZGKCooLCZGJkYmKiZGJEYmLCpGJ0YmKiRGJ0YoRiZGKUYmRi5GJkYmRihGJkYpRigtRis2JEYnRi5GJkYo

factor(series(subs(y=1/y,%),y,2));

KylJInlHNiIqJkkidEdGJCIiIi1JIkJHRiQ2JEkieEdGJCIiIUYnRiwqJiwsKihGKEYnRiZGJylGKyIiI0YnRicqKEYmRictLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YxNiNGKUYqRidGK0YnRicqJkYmRictRik2JEYsRixGJyEiIkYlRicqJkYrRidGKEYnRj9GJ0YrRj9GJy1JIk9HRjg2I0YnRjE=

eqLByneg:=collect(convert(%%%,polynom)+op(1,eqLB)+op(4,eqLB)-coeff(%,y,0),[L,B],factor);

LC4qKkkidEc2IiIiIkkieUdGJSEiIkkieEdGJUYoLUkiTEdGJTYkIiIhRi1GJkYoKipGJEYmRidGKEYpRigtRis2JEYpRi1GJkYmKipGJEYmRilGKEYnRigtSSJCR0YlNiRGLSokRidGKEYmRigqJkYkRiYtRjNGMEYmRigqKiwsKihGJEYmKUYpIiIjRiZGJ0YmRiYqKEYkRiZGKUYmKUYnRjxGJkYmKiZGJEYmRilGJkYmKiZGJEYmRidGJkYmKiZGKUYmRidGJkYoRiYtRjM2JEYpRjVGJkYpRihGPkYoRiYqKEYkRiYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0Y8NiNGM0YwRiZGJ0YoRiY=

Checking that what remains is non-negative in y

normal(series(normal(eqLB-eqLByneg),y,3));

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

... and that the negative part is indeed y-negative

normal(series(subs(y=1/y,eqLByneg),y,3));

KylJInlHNiIqJiwwKigtSSJCR0YkNiRJInhHRiQiIiEiIiJJInRHRiRGLSlGKyIiI0YtRi0qKkYwRi1GLkYtLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGMDYjRilGKkYtRitGLUYtKiYtSSJMR0YkRipGLUYuRi1GLSomRi5GLS1GPTYkRixGLEYtISIiKiZGLkYtLUYpRkBGLUZBKiZGLkYtRihGLUYtKiZGK0YtRihGLUZBRi1GK0ZBRi0sJCooI0YtRjBGLSwuKipGMEYtRi5GLUYyRi1GL0YtRi0qKkYwRi1GLkYtRihGLUYrRi1GLSooLS0mRjU2JEYwRjBGOkYqRi1GLkYtRitGLUYtKihGMEYtRi5GLUYyRi1GLSooRjBGLUYyRi1GK0YtRkEqKEYwRi1GLkYtLUYzRkBGLUZBRi1GK0ZBRi1GMC1JIk9HRjc2I0YtIiIk

This gives us an expression of 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

D2Bsol:=collect(solve(coeff(%,y),(D[2](B))(x,0)),[L,B],factor);

LCoqKCMiIiIiIiNGJUkieEc2IiEiIi1JIkxHRig2JCIiIUYtRiVGJSooRiRGJUYnRiktRis2JEYnRi1GJUYpKihGJEYlRidGKS1JIkJHRihGLEYlRiUqLEYkRiUsKComSSJ0R0YoRiUpRidGJkYlRiVGN0YlRidGKUYlRjdGKUYnRiktRjNGMEYlRik=

eqLByneg:=collect(subs((D[2](B))(x,0)=D2Bsol,eqLByneg),[L,B,L0],factor);

LC4qLCMiIiIiIiNGJUkidEc2IkYlSSJ5R0YoISIiSSJ4R0YoRiotSSJMR0YoNiQiIiFGL0YlRioqLEYkRiVGJ0YlRilGKkYrRiotRi02JEYrRi9GJUYlKixGJEYlRidGJUYpRipGK0YqLUkiQkdGKEYuRiVGJSoqRidGJUYrRipGKUYqLUY1NiRGLyokRilGKkYlRioqLEYkRiUsKiomRidGJSlGK0YmRiVGJSoqRiZGJUYnRiVGK0YlRilGJUYlRidGJUYrRipGJUYrRipGKUYqLUY1RjJGJUYqKiosLCooRidGJUY9RiVGKUYlRiUqKEYnRiVGK0YlKUYpRiZGJUYlKiZGJ0YlRitGJUYlKiZGJ0YlRilGJUYlKiZGK0YlRilGJUYqRiUtRjU2JEYrRjlGJUYrRipGREYqRiU=

eqB:=collect(1/y*subs(y=1/y,eqLByneg*2),[Ker,L,B],factor);

LC4qKEkidEc2IiIiIkkieEdGJSEiIi1JIkxHRiU2JCIiIUYsRiZGKCooRiRGJkYnRigtRio2JEYnRixGJkYmKihGJEYmRidGKC1JIkJHRiVGK0YmRiYqKiIiI0YmRiRGJkYnRigtRjI2JEYsSSJ5R0YlRiZGKCoqLCoqKEYkRiYpRidGNEYmRjdGJkYmKihGNEYmRiRGJkYnRiZGJiomRiRGJkY3RiZGJiomRidGJkY3RiZGKEYmRidGKEY3RigtRjJGL0YmRigqLEY0RiZGN0YoLCxGOkYmKihGJEYmRidGJilGN0Y0RiZGJiomRiRGJkYnRiZGJkY9RiZGPkYoRiZGJ0YoLUYyNiRGJ0Y3RiZGJg==

n:=10: normal(series(subs(B(x,y)=Bser(n), L(0,0)=subs(x=0,y=0,Lser(n)), B(0,y)=subs(x=0,Bser(n)),B(x,0)=subs(y=0,Bser(n)),B(0,0)=subs(x=0,y=0,Bser(n)), B(0,1/y)=subs(x=0,y=1/y,Bser(n)), L(0,y)=subs(x=0,Lser(n)), L(x,0)=subs(y=0,Lser(n)), L(x,y)=Lser(n), eqB),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

paper version

eqpap:=-Ker*(2*B(x,y)-B(x,0))+t*(y-1/y)*B(x,0)-2*t/x*B(0,y)+t/x*L(x,0)-t/x*L(0,0)+t/x*B(0,0);

LC4qJkkkS2VyRzYiIiIiLCYqJiIiI0YmLUkiQkdGJTYkSSJ4R0YlSSJ5R0YlRiZGJi1GKzYkRi0iIiEhIiJGJkYyKihJInRHRiVGJiwmRi5GJiokRi5GMkYyRiZGL0YmRiYqKkYpRiZGNEYmRi1GMi1GKzYkRjFGLkYmRjIqKEY0RiZGLUYyLUkiTEdGJUYwRiZGJiooRjRGJkYtRjItRjw2JEYxRjFGJkYyKihGNEYmRi1GMi1GK0Y/RiZGJg==

factor(subs(Ker=K,eqB/eqpap));

IiIi

eqB:=eqpap;

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

Finally, we check that eqB and eqL imply eqLB (which was equivalent to eqC)

Bexpr:=isolate(eqB,B(x,y)); Lexpr:=isolate(eqL,L(x,y));

Ly1JIkJHNiI2JEkieEdGJUkieUdGJSwmKigjIiIiIiIjRiwsLCooSSJ0R0YlRiwsJkYoRiwqJEYoISIiRjNGLC1GJDYkRiciIiFGLEYzKipGLUYsRjBGLEYnRjMtRiQ2JEY2RihGLEYsKihGMEYsRidGMy1JIkxHRiVGNUYsRjMqKEYwRixGJ0YzLUY8NiRGNkY2RixGLCooRjBGLEYnRjMtRiRGP0YsRjNGLEkkS2VyR0YlRjNGMyomRitGLEY0RixGLA==

Ly1JIkxHNiI2JEkieEdGJUkieUdGJSwmKigjIiIiIiIjRiwsLiooSSJ0R0YlRixGKCEiIi1GJDYkIiIhRjRGLEYxKipGJ0YxRjBGLCwmKiQpRidGLUYsRixGLEYxRiwtRiQ2JEY0RihGLEYxKipGLUYsRjBGLEYoRjEtRiQ2JEYnRjRGLEYsKihGMEYsRihGMS1JIkJHRiVGM0YsRixGLEYxKihGMEYsRihGMS1GQEY6RixGMUYsSSRLZXJHRiVGMUYxKiZGK0YsRjlGLEYs

factor(subs(Bexpr,subs(y=1/y,Bexpr),Lexpr, subs(x=1/x,Lexpr),Ker=K,eqLB));

IiIh

Pexpr:=-B(x, 0)/y+L(x, y)/x+B(x, y)/y-L(0, y)/x;

LCoqJi1JIkJHNiI2JEkieEdGJiIiISIiIkkieUdGJiEiIkYsKiYtSSJMR0YmNiRGKEYrRipGKEYsRioqJi1GJUYwRipGK0YsRioqJi1GLzYkRilGK0YqRihGLEYs

eqL:=(-2*L(x, y)+L(0, y))*Ker+t*L(0, 0)/y+t*(x^2-1)*L(0, y)/x-2*t*L(x, 0)/y-t*B(0, 0)/y+1+t*B(0, y)/y; eqB:=-Ker*(2*B(x, y)-B(x, 0))+t*(y-1/y)*B(x, 0)-2*t*B(0, y)/x+t*L(x, 0)/x-t*L(0, 0)/x+t*B(0, 0)/x;

LDAqJiwmKiYiIiMiIiItSSJMRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKihJInRHRipGJy1GKTYkRjFGMUYnRi1GLkYnKipGNEYnLCYqJClGLEYmRidGJ0YnRi5GJ0YvRidGLEYuRicqKkYmRidGNEYnLUYpNiRGLEYxRidGLUYuRi4qKEY0RictSSJCR0YqRjZGJ0YtRi5GLkYnRicqKEY0RictRkBGMEYnRi1GLkYn

LC4qJkkkS2VyRzYiIiIiLCYqJiIiI0YmLUkiQkdGJTYkSSJ4R0YlSSJ5R0YlRiZGJi1GKzYkRi0iIiEhIiJGJkYyKihJInRHRiVGJiwmRi5GJiokRi5GMkYyRiZGL0YmRiYqKkYpRiZGNEYmLUYrNiRGMUYuRiZGLUYyRjIqKEY0RiYtSSJMR0YlRjBGJkYtRjJGJiooRjRGJi1GPDYkRjFGMUYmRi1GMkYyKihGNEYmLUYrRj9GJkYtRjJGJg==

eqLbis:=collect(subs(L(x,y)=Lbis(x,y)+1/2*L(0,y),eqL),[Lbis,L,B],factor);

LDAqKCIiIyIiIi1JJUxiaXNHNiI2JEkieEdGKEkieUdGKEYlSSRLZXJHRihGJSEiIiooSSJ0R0YoRiUtSSJMR0YoNiQiIiFGM0YlRitGLUYlKixGL0YlLCZGKkYlRiVGLUYlLCZGKkYlRiVGJUYlRipGLS1GMTYkRjNGK0YlRiUqKEYvRiUtSSJCR0YoRjJGJUYrRi1GLSooRi9GJS1GO0Y4RiVGK0YtRiUqKkYkRiVGL0YlLUYxNiRGKkYzRiVGK0YtRi1GJUYl

eqBbis:=collect(subs(B(x,y)=Bbis(x,y)+1/2*B(x,0),eqB),[Bbis,L,B],factor);

LC4qKCIiIyIiIkkkS2VyRzYiRiUtSSVCYmlzR0YnNiRJInhHRidJInlHRidGJSEiIiooSSJ0R0YnRiUtSSJMR0YnNiQiIiFGM0YlRitGLUYtKihGL0YlLUYxNiRGK0YzRiVGK0YtRiUqKEYvRiUtSSJCR0YnRjJGJUYrRi1GJSoqRiRGJUYvRiUtRjk2JEYzRixGJUYrRi1GLSosRi9GJSwmRixGJUYlRi1GJSwmRixGJUYlRiVGJUYsRi0tRjlGNkYlRiU=

n:=15: normal(series(subs(Ker=K,B(0,0)=subs(x=0,y=0,Bser(n)),L(0,0)=subs(x=0,y=0,Lser(n)),B(0,y)=subs(x=0,Bser(n)),L(x,y)=Lser(n), L(x,0)=subs(y=0,Lser(n)), L(0,y)=subs(x=0,Lser(n)),L(y,0)=subs(y=0,x=y,Lser(n)), eqL),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

n:=15: normal(series(subs(Ker=K,B(0,0)=subs(x=0,y=0,Bser(n)),L(0,0)=subs(x=0,y=0,Lser(n)),B(0,y)=subs(x=0,Bser(n)),B(x,0)=subs(y=0,Bser(n)), B(x,y)=Bser(n), L(x,0)=subs(y=0,Lser(n)), L(0,y)=subs(x=0,Lser(n)),L(y,0)=subs(y=0,x=y,Lser(n)), eqB),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

<Text-field style="Heading 2" layout="Heading 2">Two independent quadrant equations</Text-field>

collect(eqL+subs({x=y,y=x},eqB),[Ker,L,B]);

LDIqJiwqKiYiIiMiIiItSSJMRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJyomRiZGJy1JIkJHRio2JEYtRixGJ0YuLUY0NiRGLUYxRidGJ0kkS2VyR0YqRidGJyoqRixGLkkidEdGKkYnLCYqJClGLEYmRidGJ0YnRi5GJ0YvRidGJyoqRiZGJ0Y6RidGLUYuLUYpNiRGLEYxRidGLkYnRicqKEY6RidGLUYuLUY0RjBGJ0YnKihGOkYnLCZGLEYnKiRGLEYuRi5GJ0Y2RidGJyoqRiZGJ0Y6RidGLUYuLUY0NiRGMUYsRidGLiooRjpGJ0YtRi4tRilGN0YnRic=

eqM:=collect(subs(L(x,y)=M(x,y)-B(y,x), L(0,y)=M(0,y)-B(y,0), L(x,0)=M(x,0)-B(0,x), L(y,0)=M(y,0)-B(0,y), %),[Ker,M,L,B],factor);

LCwqJiwmKiYiIiMiIiItSSJNRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKixJInRHRipGJywmRixGJ0YnRi5GJywmRixGJ0YnRidGJ0YsRi5GL0YnRicqKkYmRidGNEYnRi1GLi1GKTYkRixGMUYnRi5GJ0YnKihGNEYnRi1GLi1GKTYkRi1GMUYnRic=

eqMbis:=collect(subs(M(x,y)=Mbis(x,y)+1/2*M(0,y),eqM),[Mbis,M],factor);

LCwqKCIiIyIiIi1JJU1iaXNHNiI2JEkieEdGKEkieUdGKEYlSSRLZXJHRihGJSEiIiosSSJ0R0YoRiUsJkYqRiVGJUYtRiUsJkYqRiVGJUYlRiVGKkYtLUkiTUdGKDYkIiIhRitGJUYlKipGJEYlRi9GJUYrRi0tRjM2JEYqRjVGJUYtRiVGJSooRi9GJUYrRi0tRjM2JEYrRjVGJUYl

eqL-subs({x=y,y=x},eqB);

LDgqJiwmKiYiIiMiIiItSSJMRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKipGJkYnSSJ0R0YqRidGLUYuLUYpNiRGMUYxRidGJyoqRixGLkY0RicsJiokKUYsRiZGJ0YnRidGLkYnRi9GJ0YnKipGJkYnRjRGJ0YtRi4tRik2JEYsRjFGJ0YuKipGJkYnRjRGJ0YtRi4tSSJCR0YqRjZGJ0YuRidGJyooRjRGJ0YtRi4tRkBGMEYnRicqJkYyRicsJiomRiZGJy1GQDYkRi1GLEYnRictRkA2JEYtRjFGLkYnRicqKEY0RicsJkYsRicqJEYsRi5GLkYnRkhGJ0YuKipGJkYnRjRGJ0YtRi4tRkA2JEYxRixGJ0YnKihGNEYnRi1GLi1GKUZJRidGLg==

eqN:=collect(subs(L(x,y)=N(x,y)+B(y,x), L(0,y)=N(0,y)+B(y,0), L(x,0)=N(x,0)+B(0,x),L(0,0)=N(0,0)+B(0,0), L(y,0)=N(y,0)+B(0,y), %),[Ker,N,L,B],factor);

LC4qJiwmKiYiIiMiIiItSSJORzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKipGJkYnSSJ0R0YqRidGLUYuLUYpNiRGMUYxRidGJyosRjRGJywmRixGJ0YnRi5GJywmRixGJ0YnRidGJ0YsRi5GL0YnRicqKkYmRidGNEYnRi1GLi1GKTYkRixGMUYnRi5GJ0YnKihGNEYnRi1GLi1GKTYkRi1GMUYnRi4=

eqNbis:=collect(subs(N(x,y)=Nbis(x,y)+1/2*N(0,y),eqN),[Nbis,N],factor);

LC4qKCIiIyIiIkkkS2VyRzYiRiUtSSVOYmlzR0YnNiRJInhHRidJInlHRidGJSEiIioqRiRGJUkidEdGJ0YlRixGLS1JIk5HRic2JCIiIUYzRiVGJSosRi9GJSwmRitGJUYlRi1GJSwmRitGJUYlRiVGJUYrRi0tRjE2JEYzRixGJUYlKipGJEYlRi9GJUYsRi0tRjE2JEYrRjNGJUYtRiVGJSooRi9GJUYsRi0tRjE2JEYsRjNGJUYt

Mser:=proc(n) convert(normal(series(Lser(n)+subs({x=y,y=x},Bser(n)),t,n+1)),polynom): end:

Nser:=proc(n) convert(normal(series(Lser(n)-subs({x=y,y=x},Bser(n)),t,n+1)),polynom): end:

Mser(4);

LCwiIiJGIyomLCZJInlHNiJGI0kieEdGJ0YjRiNJInRHRidGI0YjKiYsKiokKUYoIiIjRiNGIyooRi5GI0YoRiNGJkYjRiMqJClGJkYuRiNGIyIiJUYjRiMpRilGLkYjRiMqJiwuKiQpRigiIiRGI0YjKihGOEYjRi1GI0YmRiNGIyooRjhGI0YoRiNGMUYjRiMqJClGJkY4RiNGIyomIiIoRiNGKEYjRiMqJiIjNUYjRiZGI0YjRiMpRilGOEYjRiMqJiw0KiQpRihGMkYjRiMqKEYyRiNGN0YjRiZGI0YjKigiIidGI0YtRiNGMUYjRiMqKEYyRiNGKEYjRjxGI0YjKiQpRiZGMkYjRiMqJiIjNkYjRi1GI0YjKigiI0JGI0YoRiNGJkYjRiMqJiIjPUYjRjFGI0YjIiNKRiNGIylGKUYyRiNGIw==

eqM:=(-2*M(x, y)+M(0, y))*Ker+t*(x-1)*(x+1)*M(0, y)/x-2*t*M(x, 0)/y+1+t*M(y, 0)/y; eqN := (-2*N(x, y)+N(0, y))*Ker+2*t*N(0, 0)/y+t*(x-1)*(x+1)*N(0, y)/x-2*t*N(x, 0)/y+1-t*N(y, 0)/y;

LCwqJiwmKiYiIiMiIiItSSJNRzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKixJInRHRipGJywmRixGJ0YnRi5GJywmRixGJ0YnRidGJ0YvRidGLEYuRicqKkYmRidGNEYnLUYpNiRGLEYxRidGLUYuRi5GJ0YnKihGNEYnLUYpNiRGLUYxRidGLUYuRic=

LC4qJiwmKiYiIiMiIiItSSJORzYiNiRJInhHRipJInlHRipGJyEiIi1GKTYkIiIhRi1GJ0YnSSRLZXJHRipGJ0YnKipGJkYnSSJ0R0YqRictRik2JEYxRjFGJ0YtRi5GJyosRjRGJywmRixGJ0YnRi5GJywmRixGJ0YnRidGJ0YvRidGLEYuRicqKkYmRidGNEYnLUYpNiRGLEYxRidGLUYuRi5GJ0YnKihGNEYnLUYpNiRGLUYxRidGLUYuRi4=

eqMbis:=-2*Mbis(x, y)*Ker+t*(x-1)*(x+1)*M(0, y)/x-2*t*M(x, 0)/y+1+t*M(y, 0)/y; eqNbis := -2*Ker*Nbis(x, y)+2*t*N(0, 0)/y+t*(x-1)*(x+1)*N(0, y)/x-2*t*N(x, 0)/y+1-t*N(y, 0)/y;

LCwqKCIiIyIiIi1JJU1iaXNHNiI2JEkieEdGKEkieUdGKEYlSSRLZXJHRihGJSEiIiosSSJ0R0YoRiUsJkYqRiVGJUYtRiUsJkYqRiVGJUYlRiUtSSJNR0YoNiQiIiFGK0YlRipGLUYlKipGJEYlRi9GJS1GMzYkRipGNUYlRitGLUYtRiVGJSooRi9GJS1GMzYkRitGNUYlRitGLUYl

LC4qKCIiIyIiIkkkS2VyRzYiRiUtSSVOYmlzR0YnNiRJInhHRidJInlHRidGJSEiIioqRiRGJUkidEdGJ0YlLUkiTkdGJzYkIiIhRjNGJUYsRi1GJSosRi9GJSwmRitGJUYlRi1GJSwmRitGJUYlRiVGJS1GMTYkRjNGLEYlRitGLUYlKipGJEYlRi9GJS1GMTYkRitGM0YlRixGLUYtRiVGJSooRi9GJS1GMTYkRixGM0YlRixGLUYt

n:=15: normal(series(subs(Ker=K,M(x,y)=Mser(n), M(x,0)=subs(y=0,Mser(n)), M(0,y)=subs(x=0,Mser(n)),M(y,0)=subs(y=0,x=y,Mser(n)), eqM),t,n+1)); normal(series(subs(Ker=K,N(x,y)=Nser(n), N(x,0)=subs(y=0,Nser(n)),N(0,0)=subs(x=0,y=0,Nser(n)), N(0,y)=subs(x=0,Nser(n)),N(y,0)=subs(y=0,x=y,Nser(n)), eqN),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

The roots of the kernel

Yens:=factor([solve(K,y)]);

NyQsJCoqIyIiIiIiI0YmSSJ4RzYiISIiSSJ0R0YpRiosKiomRitGJilGKEYnRiZGKkYrRipGKEYmKiQpKiYsKkYtRiYqKEYnRiZGK0YmRihGJkYmRitGJkYoRipGJiwqRi1GJkYzRipGK0YmRihGKkYmRiVGJkYmRiZGJiwkKipGJUYmLCpGLUYmRi9GJkYrRiZGKEYqRiZGKEYqRitGKkYq

Y0sol:=op(2,Yens);Y1sol:=op(1,Yens);

LCQqKiMiIiIiIiNGJSwqKiZJInRHNiJGJSlJInhHRipGJkYlRiUqJCkqJiwqRihGJSooRiZGJUYpRiVGLEYlRiVGKUYlRiwhIiJGJSwqRihGJUYxRjJGKUYlRixGMkYlRiRGJUYlRilGJUYsRjJGJUYsRjJGKUYyRjI=

LCQqKiMiIiIiIiNGJUkieEc2IiEiIkkidEdGKEYpLCoqJkYqRiUpRidGJkYlRilGKkYpRidGJSokKSomLCpGLEYlKihGJkYlRipGJUYnRiVGJUYqRiVGJ0YpRiUsKkYsRiVGMkYpRipGJUYnRilGJUYkRiVGJUYlRiU=

Delta:=(1-t*(x+1)*(1+1/x))*(1+t*(1-x)*(1-1/x));

KiYsJiIiIkYkKihJInRHNiJGJCwmSSJ4R0YnRiRGJEYkRiQsJkYkRiQqJEYpISIiRiRGJEYsRiQsJkYkRiQqKEYmRiQsJkYkRiRGKUYsRiQsJkYkRiRGK0YsRiRGJEYk

Y0ser:=proc(n) simplify(series(Y0sol,t,n+2),symbolic): end:

Y0ser(4);

Ky1JInRHNiIiIiJGJSomSSJ4R0YkISIiLCYqJClGJyIiI0YlRiVGJUYlRiVGLComRitGKCwoKiQpRiciIiVGJUYlKiYiIiRGJUYrRiVGJUYlRiVGJUYzKigpRidGM0YoRilGJSwoRi9GJSomIiImRiVGK0YlRiVGJUYlRiVGMS1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY4

factor(expand(Y0sol*Y1sol));

IiIi

<Text-field style="Heading 2" layout="Heading 2">Canceling the kernel: an equation between M(x,0), M(0,x) and M(0,1/x)</Text-field>

Cancel the kernel -> four equations

We use the pairs (x,Y), (1/x,Y), (Y,x) et (Y, 1/x)

f1:=collect(subs(Mbis(x,y)=0,y=Y,-eqMbis),[M]); f2:=subs(x=1/x,f1);

LCoiIiIhIiIqLEkidEc2IkYjLCZJInhHRidGI0YjRiRGIywmRilGI0YjRiNGIy1JIk1HRic2JCIiIUkiWUdGJ0YjRilGJEYkKioiIiNGI0YmRiMtRiw2JEYpRi5GI0YvRiRGIyooRiZGIy1GLDYkRi9GLkYjRi9GJEYk

LCoiIiIhIiIqLEkidEc2IkYjLCYqJEkieEdGJ0YkRiNGI0YkRiMsJkYjRiNGKUYjRiMtSSJNR0YnNiQiIiFJIllHRidGI0YqRiNGJCoqIiIjRiNGJkYjLUYtNiRGKUYvRiNGMEYkRiMqKEYmRiMtRi02JEYwRi9GI0YwRiRGJA==

f3:=collect(subs({x=y,y=x},Mbis(y,x)=0,y=Y,-eqMbis),[M]); f4:=subs(x=1/x,f3);

LCoiIiIhIiIqLEkidEc2IkYjLCZJIllHRidGI0YjRiRGIywmRilGI0YjRiNGIy1JIk1HRic2JCIiIUkieEdGJ0YjRilGJEYkKioiIiNGI0YmRiMtRiw2JEYpRi5GI0YvRiRGIyooRiZGIy1GLDYkRi9GLkYjRi9GJEYk

LCoiIiIhIiIqLEkidEc2IkYjLCZJIllHRidGI0YjRiRGIywmRilGI0YjRiNGIy1JIk1HRic2JCIiISokSSJ4R0YnRiRGI0YpRiRGJCoqIiIjRiNGJkYjLUYsNiRGKUYuRiNGMEYjRiMqKEYmRiMtRiw2JEYvRi5GI0YwRiNGJA==

indets(f1*f2*f3*f4);

PCtJIllHNiJJInRHRiRJInhHRiQtSSJNR0YkNiQiIiFGIy1GKDYkRipGJi1GKDYkRioqJEYmISIiLUYoNiRGI0YqLUYoNiRGJkYqLUYoNiRGL0Yq

We eliminate the series involving Y

MYsol:=factor(solve({f1,f2},{M(0,Y),M(Y,0)}));

PCQvLUkiTUc2IjYkIiIhSSJZR0YmLCQqLEkieEdGJiIiIiwmLUYlNiQqJEYsISIiRihGLS1GJTYkRixGKEYyRi1GKUYyLCZGLEYtRi1GMkYyLCZGLEYtRi1GLUYyRjIvLUYlNiRGKUYoKiYsKComSSJ0R0YmRi1GL0YtRi0qJkY9Ri1GM0YtRi1GKUYyRi1GPUYy

f3bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f3))),[M],factor);

LCoqLCMiIiIiIiRGJSwmSSJZRzYiRiVGJSEiIkYlLCZGKEYlRiVGJUYlRihGKi1JIk1HRik2JCIiIUkieEdGKUYlRioqKEYkRiVGMEYqLUYtNiRGMEYvRiVGJSooIyIiI0YmRiVGMEYqLUYtNiQqJEYwRipGL0YlRiUqKkYkRiUsJiomRjZGJUYoRiVGJUYwRiVGJUkidEdGKUYqRjBGKkYq

f4bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f4))),[M],factor);

LCoqLiMiIiIiIiRGJSwmSSJZRzYiRiVGJSEiIkYlLCZGKEYlRiVGJUYlSSJ4R0YpRipGKEYqLUkiTUdGKTYkIiIhKiRGLEYqRiVGKiomIyIiI0YmRiUtRi42JEYsRjBGJUYlKiZGJEYlLUYuNiRGMUYwRiVGJSoqRiRGJSwmKihGNEYlRihGJUYsRiVGJUYlRiVGJUkidEdGKUYqRixGKkYq

n:=10: normal(series(subs(M(0,x)=subs(x=0,y=x,Mser(n)),M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(x,0)=subs(y=0,Mser(n)), M(1/x,0)=subs(x=1/x,y=0,Mser(n)), Y=Y0ser(n+2),f3bis),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Eliminate M(1/x,0)

f8:=collect(-3*x*f3bis+6*f4bis,[M,x],factor);

LC4qLCwmSSJZRzYiIiIiRichIiJGJywmRiVGJ0YnRidGJ0YlRihJInhHRiZGJy1JIk1HRiY2JCIiIUYqRidGJyouIiIjRidGJEYnRilGJ0YqRihGJUYoLUYsNiRGLiokRipGKEYnRigqJiIiJEYnLUYsNiRGKkYuRidGJyomSSJ0R0YmRihGKkYnRicqKEYwRidGJUYnRjlGKEYoKihGMEYnRjlGKEYqRihGKA==

What we should remember from the above subsection

f8 := (Y-1)*(Y+1)*x*M(0, x)/Y-(2*(Y-1))*(Y+1)*M(0, 1/x)/(x*Y)+3*M(x, 0)+x/t-2*Y/t-2/(t*x);

LC4qLCwmSSJZRzYiIiIiRichIiJGJywmRiVGJ0YnRidGJ0kieEdGJkYnLUkiTUdGJjYkIiIhRipGJ0YlRihGJyouIiIjRidGJEYnRilGJy1GLDYkRi4qJEYqRihGJ0YqRihGJUYoRigqJiIiJEYnLUYsNiRGKkYuRidGJyomRipGJ0kidEdGJkYoRicqKEYwRidGJUYnRjlGKEYoKihGMEYnRipGKEY5RihGKA==

factor(expand(rationalize(subs(Y=Y0sol,Y-1/Y))));

LCQqKEkieEc2IiEiIkkidEdGJUYmKSomLCoqJkYnIiIiKUYkIiIjRixGLCooRi5GLEYnRixGJEYsRixGJ0YsRiRGJkYsLCpGK0YsRi9GJkYnRixGJEYmRiwjRixGLkYsRiY=

simplify(%/sqrt(Delta),symbolic);

LCQqJEkidEc2IiEiIkYm

Introducing De=Delta

f8form:=1/t*sqrt(De)*(x*M(0,x)-2/x*M(0,1/x))-3*M(x, 0)-x/t+2*Y/t+2/t/x;

LCwqKEkidEc2IiEiIilJI0RlR0YlIyIiIiIiI0YqLCYqJkkieEdGJUYqLUkiTUdGJTYkIiIhRi5GKkYqKihGK0YqRi5GJi1GMDYkRjIqJEYuRiZGKkYmRipGKiomIiIkRiotRjA2JEYuRjJGKkYmKiZGLkYqRiRGJkYmKihGK0YqSSJZR0YlRipGJEYmRioqKEYrRipGLkYmRiRGJkYq

n:=10: normal(series(subs(De=Delta,M(0,1/x)=subs(x=0,y=1/x,Mser(n)), M(0,x)=subs(x=0,y=x,Mser(n)),M(x,0)=subs(y=0,Mser(n)), Y=Y0ser(n+2),f8form),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

<Text-field style="Heading 2" layout="Heading 2">An equation between M(0,x) and M(0,1/x) (or S(x) and S(1/x))</Text-field>

Introduce the series R and S

Rser:=proc(n) collect(t*subs(y=0,Mser(n-1)),t,normal): end:

Sser:=proc(n) collect(subs(x=0,y=x,Mser(n)*y*t),t); end:

Rser(4);Sser(4);

LCoqJiwmKiQpSSJ4RzYiIiIkIiIiRioqJiIiKEYqRidGKkYqRiopSSJ0R0YoIiIlRipGKiomLCYqJClGJyIiI0YqRipGL0YqRiopRi5GKUYqRioqJkYnRiopRi5GNEYqRipGLkYq

LCwqKEkieEc2IiIiIiwoKiQpRiQiIiVGJkYmKiYiIz1GJilGJCIiI0YmRiYiI0pGJkYmKUkidEdGJSIiJkYmRiYqKEYkRiYsJiokKUYkIiIkRiZGJiomIiM1RiZGJEYmRiZGJilGMUYqRiZGJiooLCYqJEYtRiZGJkYqRiZGJkYkRiYpRjFGN0YmRiYqJkYtRiYpRjFGLkYmRiYqJkYkRiZGMUYmRiY=

f13:=collect(subs(M(0,x)=S(x)/t/x,M(0,1/x)=x/t*S(1/x), M(x,0)=R(x)/t,Y=(1-t*(x+1/x)-sqrt(De))/2/t,t^2*f8form),[De,R],expand);

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op(1,f13);

KiYsKC1JIlNHNiI2I0kieEdGJiIiIiomIiIjRiktRiU2IyokRighIiJGKUYvRilGL0YpKUkjRGVHRiYjRilGK0Yp

u0:=op(1,f13)^2-subs(De=0,f13)^2;

LCYqJiksKC1JIlNHNiI2I0kieEdGKCIiIiomIiIjRistRic2IyokRiohIiJGK0YxRitGMUYtRitJI0RlR0YoRitGKyokKSwqKigiIiRGK0kidEdGKEYrLUkiUkdGKEYpRitGMSooRi1GK0YqRitGOEYrRjFGK0YrKiZGOEYrRipGMUYrRi1GK0Yx

u1:=collect(u0,[S,R],distributed,expand);

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u1expl:=collect(subs(De=Delta,u0),[S,R],distributed,expand);

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Extract the negative part in x (which must be zero). Recall that S(0)=0

series(Sser(3),t,4);

KytJInRHNiJJInhHRiQiIiIqJClGJSIiI0YmRikqJiwmRidGJiIiJUYmRiZGJUYmIiIkLUkiT0clKnByb3RlY3RlZEc2I0YmRiw=

Also, R(0) and D(S)(0) are equal by definition.

subs(x=0,Rser(n));coeff(Sser(n),x,1);

LCwqJiIlJ0ckIiIiKUkidEc2IiIiKkYlRiUqJiIkLCRGJSlGJyIiKEYlRiUqJiIjSkYlKUYnIiImRiVGJSomIiIlRiUpRiciIiRGJUYlRidGJQ==

LC4qJiImaSZRIiIiKUkidEc2IiIjNkYlRiUqJiIlJ0ckRiUpRiciIipGJUYlKiYiJCwkRiUpRiciIihGJUYlKiYiI0pGJSlGJyIiJkYlRiUqJiIiJUYlKUYnIiIkRiVGJUYnRiU=

The contribution of the series in x

subs(S(0)=0,R(0)=D(S)(0),series(subs(S(1/x)=0,u1expl),x,2));

KydJInhHNiIsJiooIiIlIiIiKUkidEdGJCIiI0YoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRihGKComRidGKEYqRighIiJGNy1JIk9HRjA2I0YoRjU=

Some parts of the series in 1/x must be subtracted

op(4,u1expl)+op(5,u1expl);

LCYqJiwuIiIlIiIiKigiIilGJkkidEc2IkYmSSJ4R0YqISIiRiwqKEYoRiZGK0YmRilGJkYsKiZGKEYmKUYpIiIjRiZGLCooRiVGJilGK0YwRiZGL0YmRiYqKEYlRiZGL0YmRjJGLEYmRiYpLUkiU0dGKjYjKiRGK0YsRjBGJkYmKiZGJEYmRjVGJkYm

factor(subs(S(0)=0,series(subs(x=1/x,%),x,3)));

KylJInhHNiIsJCooIiIlIiIiKUkidEdGJCIiI0YoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRihGKCEiIiwkKihGK0YoRipGKCwoKihGK0YoKUYsRitGKEYqRihGKComLS0tSSNAQEdGLzYkRi5GK0YyRjRGKEYqRihGKComRidGKEYsRihGNkYoRihGNS1JIk9HRjA2I0YoRig=

The negative part

u2:=convert(%%%,polynom)-4*Sneg(1/x)+%%-subs(x=1/x,convert(%,polynom));

LC4qJiwmKigiIiUiIiIpSSJ0RzYiIiIjRictLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjSSJTR0YqNiMiIiFGJ0YnKiZGJkYnRilGJyEiIkYnSSJ4R0YqRjdGJyomRiZGJy1JJVNuZWdHRio2IyokRjhGN0YnRjcqJiwuRiZGJyooIiIpRidGKUYnRjhGN0Y3KihGQUYnRjhGJ0YpRidGNyomRkFGJ0YoRidGNyooRiZGJylGOEYrRidGKEYnRicqKEYmRidGKEYnRkVGN0YnRicpLUYzRjxGK0YnRicqJkY/RidGSEYnRicqKkYmRidGKEYnRixGJ0Y4RidGNyooRitGJ0YpRicsKCooRitGJylGLEYrRidGKUYnRicqJi0tLUkjQEBHRi82JEYuRitGMkY0RidGKUYnRicqJkYmRidGLEYnRjdGJ0Y3

let us check: this must be non-negative

n:=10: expand(series(subs(De=Delta,Sneg(1/x)=Delta*S(x)*S(1/x),S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,D(S)(0)=subs(x=0,Rser(n)),(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u2),t,n));

KzVJInRHNiIhIiUiIiMsJComIiIlIiIiSSJ4R0YkRipGKiIiJCEjN0YpLCQqJiIjQ0YqRitGKkYqIiImLCYqJiIiKUYqKUYrRiZGKkYqIiMjKiEiIiIiJywmKiZGNEYqKUYrRixGKkYqKiYiJDMjRipGK0YqRioiIigsKComRjRGKilGK0YpRipGKiomIiQ/IkYqRjVGKkYqIiQjKilGN0Y0LCgqJkY0RiopRitGMUYqRioqJiIkayJGKkY7RipGKiomIiU7QEYqRitGKkYqIiIqLUkiT0clKnByb3RlY3RlZEc2I0YqIiM1

and this must be negative

n:=14: expand(series(subs(De=Delta,Sneg(1/x)=0,S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,D(S)(0)=subs(x=0,Rser(n)), (D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u2),t,n));

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

u2form:=(4*t^2*(D(S))(0)-4*t)/x-4*Sneg(1/x)+4*De*S(1/x)^2+4*De*S(1/x)-4*t^2*(D(S))(0)*x-2*t*(2*(D(S))(0)^2*t+((D@@2)(S))(0)*t-4*(D(S))(0));

LC4qJiwmKigiIiUiIiIpSSJ0RzYiIiIjRictLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjSSJTR0YqNiMiIiFGJ0YnKiZGJkYnRilGJyEiIkYnSSJ4R0YqRjdGJyomRiZGJy1JJVNuZWdHRio2IyokRjhGN0YnRjcqKEYmRidJI0RlR0YqRicpLUYzRjxGK0YnRicqKEYmRidGP0YnRkFGJ0YnKipGJkYnRihGJ0YsRidGOEYnRjcqKEYrRidGKUYnLCgqKEYrRicpRixGK0YnRilGJ0YnKiYtLS1JI0BAR0YvNiRGLkYrRjJGNEYnRilGJ0YnKiZGJkYnRixGJ0Y3RidGNw==

u3:=map(Z->factor(Z/4),u2);

LC4qKEkidEc2IiIiIiwmKiZGJEYmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kiU0dGJTYjIiIhRiZGJkYmISIiRiZJInhHRiVGM0YmLUklU25lZ0dGJTYjKiRGNEYzRjMqKiwqKiZGJEYmKUY0IiIjRiZGJiooRj1GJkYkRiZGNEYmRiZGJEYmRjRGM0YmLCpGO0YmRj5GM0YkRiZGNEYzRiZGPEYzKS1GMEY3Rj1GJkYmKipGOkYmRj9GJkY8RjNGQUYmRiYqKClGJEY9RiZGKUYmRjRGJkYzKigjRiZGPUYmRiRGJiwoKihGPUYmKUYpRj1GJkYkRiZGJiomLS0tSSNAQEdGLDYkRitGPUYvRjFGJkYkRiZGJiomIiIlRiZGKUYmRjNGJkYz

u3form:=S(1/x)*(S(1/x)+1)*De-Sneg(1/x)-F0*x-F1-F2/x;

LCwqKC1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCZGJEYrRitGK0YrSSNEZUdGJkYrRistSSVTbmVnR0YmRidGKiomSSNGMEdGJkYrRilGK0YqSSNGMUdGJkYqKiZJI0YyR0YmRitGKUYqRio=

F0expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form/4),x,1); F1expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form/4),x,0); F2expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form/4),x,-1);

KiYpSSJ0RzYiIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kiU0dGJTYjIiIhRic=

LCQqKCMiIiIiIiNGJUkidEc2IkYlLCgqKEYmRiUpLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0kiU0dGKDYjIiIhRiZGJUYnRiVGJSomLS0tSSNAQEdGLzYkRi5GJkYyRjRGJUYnRiVGJSomIiIlRiVGLEYlISIiRiVGJQ==

LCYqJilJInRHNiIiIiMiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSJTR0YmNiMiIiFGKCEiIkYlRig=

We now reconstruct Delta S(x) S(1/x)

u5:=collect((factor( De*S(x)*S(1/x)-solve(u3form,Sneg(1/x))-P0-subs(x=1/x,solve(u3form,Sneg(1/x)) ))),[De,F0,F1,F2],expand);

LCwqJiwsKiQpLUkiU0c2IjYjSSJ4R0YpIiIjIiIiISIiKiZGJ0YtLUYoNiMqJEYrRi5GLUYtKiQpRjBGLEYtRi5GJ0YuRjBGLkYtSSNEZUdGKUYtRi0qJiwmRitGLUYyRi1GLUkjRjBHRilGLUYtKiZGLEYtSSNGMUdGKUYtRi0qJkY3Ri1JI0YyR0YpRi1GLUkjUDBHRilGLg==

Let us check

n:=15: expand(series(subs(De=Delta,F0=F0expr,F1=F1expr,F2=F2expr,S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)),D(S)(0)=subs(x=0,Rser(n)) ,(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0), u5),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM6

What to remember from the previous sub-section

Rser:=proc (n) collect(t*subs(y = 0, Mser(n-1)), t, normal) end proc:

Sser:=proc(n) collect(subs(x=0,y=x,Mser(n)*y*t),t) end proc:

f13:=(S(x)-2*S(1/x)-1)*sqrt(De)-3*t*R(x)-2*x*t+1+t/x;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjAqKEYsRipGKUYqRjZGKkYwRipGKiomRjZGKkYpRjBGKg==

u5:=(-S(x)^2+S(x)*S(1/x)-S(1/x)^2-S(x)-S(1/x))*De+(x+1/x)*F0+2*F1+(x+1/x)*F2-P0;

F0expr:=t^2*(D(S))(0); F1expr:=(1/2)*t*(2*(D(S))(0)^2*t+((D@@2)(S))(0)*t-4*(D(S))(0)); F2expr:=-t^2*(D(S))(0)+t;

KiYpSSJ0RzYiIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kiU0dGJTYjIiIhRic=

LCQqKCMiIiIiIiNGJUkidEc2IkYlLCgqKEYmRiUpLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0kiU0dGKDYjIiIhRiZGJUYnRiVGJSomLS0tSSNAQEdGLzYkRi5GJkYyRjRGJUYnRiVGJSomIiIlRiVGLEYlISIiRiVGJQ==

LCYqJilJInRHNiIiIiMiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSJTR0YmNiMiIiFGKCEiIkYlRig=

<Text-field style="Heading 2" layout="Heading 2">An equation for M(0,x) (or S(x))</Text-field>

The right multiplication

u11:=collect(expand(u5*(2+S(x)+S(1/x))),[De,S],distributed,expand);

LD4qJiIiIyIiIkkjUDBHNiJGJSEiIiooRiRGJUkjRjJHRidGJUkieEdGJ0YoRiUqKEYkRiVJI0YwR0YnRiVGK0YlRiUqJiIiJUYlSSNGMUdGJ0YlRiUqKEYkRiVGLUYlRitGKEYlKihGJEYlRipGJUYrRiVGJSomLC4qJkYqRiVGK0YlRiUqJkYtRiVGK0YoRiUqJkYtRiVGK0YlRiUqJkYqRiVGK0YoRiUqJkYkRiVGMEYlRiVGJkYoRiUtSSJTR0YnNiNGK0YlRiUqJkY0RiUtRjs2IyokRitGKEYlRiUqKCIiJEYlSSNEZUdGJ0YlKUY6RiRGJUYoKihGJEYlRkNGJUY6RiVGKCooRkJGJUZDRiUpRj5GJEYlRigqKEYkRiVGQ0YlRj5GJUYoKiZGQ0YlKUY6RkJGJUYoKiZGQ0YlKUY+RkJGJUYo

factor(u11/subs(x=1/x,u11));

IiIi

Extract the non-negative part. We first look at the terms other than S(1/x)

normal(subs(S(0)=0,series(subs(S(1/x)=0,De=Delta,u11),x,2)));

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The terms involving S(1/x)

u11-subs(S(1/x)=0,u11);

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normal(subs(S(0)=0,series(subs(x=1/x,De=Delta,%),x,3)));

KylJInhHNiIsJCooIiIjIiIiKUkidEdGJEYnRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJTR0YkNiMiIiFGKCEiIkY1LCwqKCIiJEYoRilGKClGK0YnRihGNSomRilGKC0tLUkjQEBHRi42JEYtRidGMUYzRihGNSomRitGKEkjRjBHRiRGKEYoKiZGK0YoSSNGMkdGJEYoRigqKCIiJUYoRipGKEYrRihGKEY0LUkiT0dGLzYjRihGKA==

eqSp:=subs(S(1/x)=0,u11)-convert(%%%,polynom)+subs(x=1/x,convert(%,polynom));

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

Let us check this equation

n:=16: normal(series(subs(S(x)=Sser(n),De=Delta,F0=F0expr, F1=F1expr,F2=F2expr,D(S)(0)=subs(x=0,Rser(n)),((D@@2)(S))(0)=subs(x=0,diff(Sser(n),x$2)),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0), eqSp),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM8

How to rewrite this ? At the moment we have three unknown series in t, namely D(S)(0), 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

eqSp1:=collect(subs(F1=F1expr,F2=F2expr,F0=F0expr,eqSp),[De,S(x),x],factor);

LDIqJiwoKiQpLUkiU0c2IjYjSSJ4R0YpIiIkIiIiISIiKiZGLEYtKUYnIiIjRi1GLiomRjFGLUYnRi1GLkYtSSNEZUdGKUYtRi0qJiwuKiZGK0YtSSJ0R0YpRi1GLSooRjFGLSlGN0YxRi0pLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2I0YoNiMiIiFGMUYtRi0qJkY5Ri0tLS1JI0BAR0Y+NiRGPUYxRkFGQkYtRi0qKCIiJUYtRjdGLUY7Ri1GLkkjUDBHRilGLiomRjdGLUYrRi5GLUYtRidGLUYtKipGMUYtRjdGLSwmKiZGN0YtRjtGLUYtRi1GLkYtRitGLUYuKiZGOUYtRjpGLUYtRkRGLSooRixGLUY3Ri1GO0YtRi4qJkYxRi1GTEYtRi4qKkYxRi1GOUYtRjtGLUYrRi5GLQ==

But we can express P0 in terms of the other two

factor(subs(S(0)=0,series(subs(De=Delta,eqSp1),x,3)));

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P0sol:=factor(solve(op(1,%),P0));

LCQqKC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJIlNHRio2IyIiISIiIkkidEdGKkYvLCYqJkYwRi9GJEYvRi9GLyEiIkYvRjM=

eqSp2:=collect(subs(P0=P0sol,eqSp1),[De,S(x),x],factor);

LCwqJiwoKiQpLUkiU0c2IjYjSSJ4R0YpIiIkIiIiISIiKiZGLEYtKUYnIiIjRi1GLiomRjFGLUYnRi1GLkYtSSNEZUdGKUYtRi0qJiwoKiZGK0YtSSJ0R0YpRi1GLSomRjdGLSwoKihGLEYtKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGKDYjIiIhRjFGLUY3Ri1GLSomLS0tSSNAQEdGPzYkRj5GMUZCRkNGLUY3Ri1GLSomIiImRi1GPEYtRi5GLUYtKiZGN0YtRitGLkYtRi1GJ0YtRi0qKkYxRi1GN0YtLCYqJkY3Ri1GPEYtRi1GLUYuRi1GK0YtRi5GOEYtKipGMUYtKUY3RjFGLUY8Ri1GK0YuRi0=

What needs to be remembered from this subsection

eqSp2:=(-S(x)^3-3*S(x)^2-2*S(x))*De+(x*t+t*(3*(D(S))(0)^2*t+((D@@2)(S))(0)*t-5*(D(S))(0))+t/x)*S(x)-2*t*(t*(D(S))(0)-1)*x+t*(3*(D(S))(0)^2*t+((D@@2)(S))(0)*t-5*(D(S))(0))+2*t^2*(D(S))(0)/x;

n:=10: normal(series(subs(De=Delta,S(x)=Sser(n), D(S)(0)=subs(x=0,Rser(n)), ((D@@2)(S))(0)=subs(x=0,diff(Sser(n),x$2)), eqSp2),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

<Text-field style="Heading 2" layout="Heading 2">The (generalized) quadratic method: algebraicity (for M)</Text-field>

eqS:=map(factor,subs(De=Delta,S(x)=Sx,D(S)(0)=S1,((D@@2)(S))(0)=2*S2,eqSp2));

LCwqLkkjU3hHNiIiIiIsJkYkRiYiIiNGJkYmLCZGJEYmRiZGJkYmLCoqJkkidEdGJUYmKUkieEdGJUYoRiZGJiooRihGJkYsRiZGLkYmRiZGLEYmRi4hIiJGJiwqRitGJkYvRjBGLEYmRi5GMEYmRi1GMEYwKipGLEYmLCwqKiIiJEYmKUkjUzFHRiVGKEYmRixGJkYuRiZGJioqRihGJkkjUzJHRiVGJkYsRiZGLkYmRiYqKCIiJkYmRjdGJkYuRiZGMCokRi1GJkYmRiZGJkYmRiRGJkYuRjBGJioqRihGJkYsRiYsJiomRjdGJkYsRiZGJkYmRjBGJkYuRiZGMComRixGJiwoKihGNUYmRjZGJkYsRiZGJiooRihGJkY5RiZGLEYmRiYqJkY7RiZGN0YmRjBGJkYmKipGKEYmKUYsRihGJkY3RiZGLkYwRiY=

Differentiating with respect to x_0

eqSS:=diff(eqS,Sx);

LCoqLCwmSSNTeEc2IiIiIiIiI0YnRicsJkYlRidGJ0YnRicsKiomSSJ0R0YmRicpSSJ4R0YmRihGJ0YnKihGKEYnRixGJ0YuRidGJ0YsRidGLiEiIkYnLCpGK0YnRi9GMEYsRidGLkYwRidGLUYwRjAqLEYlRidGKUYnRipGJ0YxRidGLUYwRjAqLEYlRidGJEYnRipGJ0YxRidGLUYwRjAqKEYsRicsLCoqIiIkRicpSSNTMUdGJkYoRidGLEYnRi5GJ0YnKipGKEYnSSNTMkdGJkYnRixGJ0YuRidGJyooIiImRidGOUYnRi5GJ0YwKiRGLUYnRidGJ0YnRidGLkYwRic=

We want to know how many series X(t) cancel eqSS. recall that S(x) is a multiple of t and x

op(Sser);

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJXN1YnNHRik2JS9JInhHRiUiIiEvSSJ5R0YlRjAqKC1JJU1zZXJHRiVGIyIiIkYzRjdJInRHRiVGN0Y4RiVGJUYl

numer(factor(subs(Sx=t*x*Mx,eqSS)));

LEwqKiIiJCIiIilJI014RzYiIiIjRiUpSSJ0R0YoIiIlRiUpSSJ4R0YoIiInRiUhIiIqKkYvRiVGJkYlRipGJSlGLkYsRiVGJSoqRi9GJUYmRiUpRitGJEYlKUYuIiImRiVGJSoqRi9GJUYnRiVGNEYlRjVGJUYwKipGJEYlRiZGJUYqRiUpRi5GKUYlRjAqKkYvRiVGJkYlRjRGJSlGLkYkRiVGJSoqRiRGJUYmRiUpRitGKUYlRjJGJUYwKioiIzdGJUYnRiVGNEYlRjtGJUYlKipGP0YlRidGJUY9RiVGMkYlRiUqKkYkRiUpSSNTMUdGKEYpRiVGPUYlRjlGJUYlKihGKUYlRj1GJUYyRiVGMCoqRi9GJUYnRiVGNEYlRi5GJUYwKipGP0YlRidGJUY9RiVGOUYlRiUqKkYvRiVGJ0YlRitGJUY7RiVGMCoqRilGJUkjUzJHRihGJUY9RiVGOUYlRiUqKkY2RiVGQ0YlRitGJUY5RiVGMCooRixGJUY9RiVGOUYlRiUqKEY2RiVGK0YlRjtGJUYlKiZGKUYlRj1GJUYwKihGNkYlRitGJUYuRiVGJSomRilGJUY5RiVGMA==

We thus have two roots X0 and X1 to eqSS

subs(t=0,%);

LCQqJiIiIyIiIilJInhHNiJGJEYlISIi

Let us determine their first coefficients

with(gfun):

n:=20:expr:=normal(series(subs(Sx=Sser(n), S1=subs(x=0,Rser(n)), S2=1/2*subs(x=0,diff(Sser(n),x$2)),eqSS),t,n+1)):

Xisol:=algeqtoseries(numer(convert(expr,polynom)),t,x,10,true);

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

allvalues(Xisol);

NiQ3IysvSSJ0RzYiIyIiIiIiI0YoI0YoIiIpIiIkIyIjRCIjOyIiJiMiJSZvIiIkRyIiIigjIiYyWyQiJGMjIiIqLUkiT0clKnByb3RlY3RlZEc2I0YoIiM2NyMrL0YlRilGKEYrRiwiI2tGMCIkUydGNCIlb3JGOEY5Rj0=

X1ser:=(1/2)*t+(1/8)*t^3+(25/16)*t^5+(1685/128)*t^7+(34807/256)*t^9+O(t^11); X0ser:=2*t+8*t^3+64*t^5+640*t^7+7168*t^9+O(t^11);

LC4qJiMiIiIiIiNGJUkidEc2IkYlRiUqJiNGJSIiKUYlKUYnIiIkRiVGJSomIyIjRCIjO0YlKUYnIiImRiVGJSomIyIlJm8iIiRHIkYlKUYnIiIoRiVGJSomIyImMlskIiRjI0YlKUYnIiIqRiVGJS1JIk9HJSpwcm90ZWN0ZWRHNiMqJClGJyIjNkYlRiU=

LC4qJiIiIyIiIkkidEc2IkYlRiUqJiIiKUYlKUYmIiIkRiVGJSomIiNrRiUpRiYiIiZGJUYlKiYiJFMnRiUpRiYiIihGJUYlKiYiJW9yRiUpRiYiIipGJUYlLUkiT0clKnByb3RlY3RlZEc2IyokKUYmIiM2RiVGJQ==

One root seems to be Catalan-ish

expand(1/2/t*add(binomial(2*n,n)/(n+1)*(4*t^2)^(n+1),n=0..6));

LDAqJiIoVzgzIiIiIilJInRHNiIiIzhGJUYlKiYiJjtnKUYlKUYnIiM2RiVGJSomIiVvckYlKUYnIiIqRiVGJSomIiRTJ0YlKUYnIiIoRiVGJSomIiNrRiUpRiciIiZGJUYlKiYiIilGJSlGJyIiJEYlRiUqJiIiI0YlRidGJUYl

X0rac:=(1-sqrt(1-16*t^2))/4/t;

LCQqKCMiIiIiIiVGJSwmRiVGJSokKSwmKiYiIztGJSlJInRHNiIiIiNGJSEiIkYlRiUjRiVGMEYlRjFGJUYuRjFGJQ==

series(X0rac,t,20);

KzdJInRHNiIiIiMiIiIiIikiIiQiI2siIiYiJFMnIiIoIiVvciIiKiImO2cpIiM2IihXODMiIiM4IilzdTA5IiM6IipnSFYoPSIjPC1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjPg==

Let us check this conjecture

n:=20: normal(series(subs(Sx=Sser(n), S1=coeff(Rser(n),x,0),S2=1/2*subs(x=0,diff(Sser(n),x$2)),x=X0rac,eqSS),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM+

algX0:=subs(X0(t)=X0,op(1,seriestoalgeq(series(X0rac,t,20),X0(t))));

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

tX0:=factor(solve(%,t));

LCQqKCMiIiIiIiNGJUkjWDBHNiJGJSwmKiQpRidGJkYlRiVGJUYlISIiRiU=

Differentiating w.r.t. x

eqSx:=collect(diff(eqS,x),[Sx,S1,S2],factor);

LCwqMCIiIyIiIkkidEc2IkYlLCZJInhHRidGJUYlISIiRiUsJkYpRiVGJUYlRiUsKComRiZGJSlGKUYkRiVGJUYmRiVGKUYqRiUpRikiIiRGKilJI1N4R0YnRjBGJUYqKjAiIidGJUYmRiVGKEYlRitGJUYsRiVGL0YqKUYyRiRGJUYqKi5GJkYlRihGJUYrRiUsKCooIiIlRiVGJkYlRi5GJUYlKiZGOUYlRiZGJUYlKiYiIiZGJUYpRiVGKkYlRi9GKkYyRiVGKiosRiRGJSlGJkYkRiUsJiokRi5GJUYlRiVGJUYlRi5GKkkjUzFHRidGJUYqKiZGJEYlRiZGJUYl

<Text-field style="Heading 3" layout="Heading 3">Building an explanation for the Catalan solution X0</Text-field>

factor(coeff(eqS,R0));factor(coeff(eqSS,R0));factor(coeff(eqSx,R0)); factor(coeff(eqS,S2));factor(coeff(eqSS,S2));factor(coeff(eqSx,S2));

LCQqKEkidEc2IiIiIiwqKigiIiNGJkYkRiYpSSJ4R0YlRilGJkYmKigiIiZGJkkjU3hHRiVGJkYrRiZGJiomRilGJkYkRiYhIiIqJkYtRiZGK0YmRiZGJkYrRjBGMA==

LCQqJiIiJiIiIkkidEc2IkYlISIi

LCQqKiIiIyIiIilJInRHNiJGJEYlLCYqJClJInhHRihGJEYlRiVGJUYlRiVGKyEiIkYt

LCQqKCIiIyIiIilJInRHNiJGJEYlLCZJI1N4R0YoRiVGJUYlRiVGJQ==

LCQqJiIiIyIiIilJInRHNiJGJEYlRiU=

IiIh

We eliminate R0 and S2; first S2

r1:=collect(factor(expand(eqS-(Sx+1)*eqSS)),R0,factor);

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

eqSx;

Then R0 -- and there appears the fact algX0

factor(r1*(x^2+1)/x-eqSx*(x-1)*(x+1));

KiosKCooIiIjIiIiSSJ0RzYiRiYpSSJ4R0YoRiVGJkYmKiZGJUYmRidGJkYmRiohIiJGJiwmSSNTeEdGKEYmRiZGJkYmLEAqKkYlRiYpRi5GJUYmRidGJilGKiIiJUYmRiYqKkYzRiZGLkYmRidGJkYyRiZGJioqRjNGJkYxRiZGJ0YmRilGJkYsKihGJUYmRjFGJilGKiIiJEYmRiwqJkYnRiZGMkYmRiYqKiIiKUYmRi5GJkYnRiZGKUYmRiwqKEYzRiZGLkYmRjdGJkYsKihGJUYmRjFGJkYnRiZGJiooRiVGJkYxRiZGKkYmRixGJEYsKiZGJUYmRjdGJkYsKihGM0YmRi5GJkYnRiZGJiooRjNGJkYuRiZGKkYmRixGJ0YmKiZGJUYmRipGJkYsRiZGN0Ys

algX0;

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

The complete expression

collect((eqSf-(Sx+1)*eqSSf)*(x^2+1)/x-eqSxf*(x-1)*(x+1),[eqSf,eqSSf,eqSxf],factor);

LCgqKEkieEc2IiEiIiwmKiQpRiQiIiMiIiJGK0YrRitGK0klZXFTZkdGJUYrRisqKiwmSSNTeEdGJUYrRitGK0YrSSZlcVNTZkdGJUYrRidGK0YkRiZGJiooSSZlcVN4ZkdGJUYrLCZGJEYrRitGJkYrLCZGJEYrRitGK0YrRiY=

The explanation for X0

factor((eqS-(Sx+1)*eqSS)*(x^2+1)/x-eqSx*(x-1)*(x+1));

Why X0 is a solution: this linear combination admits algXO as a factor

expr:=factor((eqS-(Sx+1)*eqSS)*(x^2+1)/x-eqSx*(x-1)*(x+1));

algX0;

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

fact:=x^3*expr;

KigsKCooIiIjIiIiSSJ0RzYiRiYpSSJ4R0YoRiVGJkYmKiZGJUYmRidGJkYmRiohIiJGJiwmSSNTeEdGKEYmRiZGJkYmLEAqKkYlRiYpRi5GJUYmRidGJilGKiIiJUYmRiYqKkYzRiZGLkYmRidGJkYyRiZGJioqRjNGJkYxRiZGJ0YmRilGJkYsKihGJUYmRjFGJilGKiIiJEYmRiwqJkYnRiZGMkYmRiYqKiIiKUYmRi5GJkYnRiZGKUYmRiwqKEYzRiZGLkYmRjdGJkYsKihGJUYmRjFGJkYnRiZGJiooRiVGJkYxRiZGKkYmRixGJEYsKiZGJUYmRjdGJkYsKihGM0YmRi5GJkYnRiZGJiooRjNGJkYuRiZGKkYmRixGJ0YmKiZGJUYmRipGJkYsRiY=

Let us check which X_i cancels which term

normal(series(subs(x=X0ser,op(1,fact)),t)); normal(series(subs(x=X1ser,op(1,fact)),t));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

KytJInRHNiIjIiIkIiIjIiIiI0YmIiIpRiYjISNAIiM7IiImLUkiT0clKnByb3RlY3RlZEc2I0YoIiIo

normal(series(subs(Sx=Sser(n),x=X0ser,op(2,fact)),t)); normal(series(subs(Sx=Sser(n),x=X1ser,op(2,fact)),t));

KytJInRHNiIiIiIiIiEiIiNGJyIjPyIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

KytJInRHNiIiIiIiIiEjRiUiIiNGKCMiIz4iIikiIiUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIic=

normal(series(subs(Sx=Sser(n),x=X0ser,op(3,fact)),t)); normal(series(subs(Sx=Sser(n),x=X1ser,op(3,fact)),t));

KytJInRHNiIhIiQiIiIhI1siIiQhJFsnIiImLUkiT0clKnByb3RlY3RlZEc2I0YmIiIo

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

Here is the term cancelled by X1

eqsupX:=op(3,fact);

LEAqKiIiIyIiIilJI1N4RzYiRiRGJUkidEdGKEYlKUkieEdGKCIiJUYlRiUqKkYsRiVGJ0YlRilGJUYqRiVGJSoqRixGJUYmRiVGKUYlKUYrRiRGJSEiIiooRiRGJUYmRiUpRisiIiRGJUYwKiZGKUYlRipGJUYlKioiIilGJUYnRiVGKUYlRi9GJUYwKihGLEYlRidGJUYyRiVGMCooRiRGJUYmRiVGKUYlRiUqKEYkRiVGJkYlRitGJUYwKihGJEYlRilGJUYvRiVGMComRiRGJUYyRiVGMCooRixGJUYnRiVGKUYlRiUqKEYsRiVGJ0YlRitGJUYwRilGJSomRiRGJUYrRiVGMA==

eqpap := (2*t+x+1/x-t*(x^2+1/x^2))*Sx*(2+Sx)+t+(x+1/x)-t*(x^2+1/x^2)/2;

LCwqKCwqKiYiIiMiIiJJInRHNiJGJ0YnSSJ4R0YpRicqJEYqISIiRicqJkYoRicsJiokKUYqRiZGJ0YnKiRGMEYsRidGJ0YsRidJI1N4R0YpRicsJkYyRidGJkYnRidGJ0YoRidGKkYnRitGJyooI0YnRiZGJ0YoRidGLkYnRiw=

factor(eqsupX/eqpap);

LCQqJiIiIyIiIilJInhHNiJGJEYlISIi

collect(eqpap

LC4iIiJGIyooIiIjRiNJInRHNiJGI0kieEdGJyEiIkYpKihGJUYjRihGI0YmRiNGKSomRiVGIylGJkYlRiNGKSomKUYoRiVGI0YsRiNGIyomRixGI0YuRilGIw==

We now want to find equations for S1 and S2 by eliminating the X_i's and S(X_i)'s

Elimination of Sx (for x=X0). The term e1 is the discriminant with respect to Sx

e1:=numer(factor(resultant(eqS,eqSS,Sx))):nops(e1);

IiIk

op(1,e1); op(2,e1);

KiQpLCoqJkkidEc2IiIiIilJInhHRiciIiNGKEYoKihGK0YoRiZGKEYqRihGKEYmRihGKiEiIkYrRig=

KiQpLCoqJkkidEc2IiIiIilJInhHRiciIiNGKEYoKihGK0YoRiZGKEYqRighIiJGJkYoRipGLUYrRig=

algX0;

LCgqKCIiIyIiIilJI1gwRzYiRiRGJUkidEdGKEYlRiVGJyEiIiomRiRGJUYpRiVGJQ==

The third factor vanishes

nops(op(3,e1));

IiQ1Ig==

e2:=op(3,e1):

and then of x

e3:=numer(factor(resultant(e2,op(1,fact),x)));

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

A first relation between S1 and S2

eqS1S21:=op(1,op(3,e3)):

We now work similarly with X1

e4:=numer(factor(resultant(eqS,op(3,fact),Sx)));nops(e4);

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

IiIm

e5:=op(5,e4):

Elimination of x

e6:=numer(factor(resultant(e1,e5,x))):nops(e6);

IiIn

Selecting the correct factor

n:=7: for i from 1 to 6 do normal(series(subs(S1=coeff(Rser(n),x,0),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(i,e6)),t,n+1));od;

KyVJInRHNiIiNTs7YjRQMlduVz0iIiE=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiRXIg==

KzNJInRHNiIiIiIiIiEhI0MiIiMiIzsiIiQiJCc+IiIlISRTIyIiJiEkJ3AiIiciJEcqIiIoLUkiT0clKnByb3RlY3RlZEc2I0YlIiIp

KzNJInRHNiIiIiIiIiEhI0MiIiMhIzsiIiQiJCc+IiIlIiRTIyIiJiEkJ3AiIichJEcqIiIoLUkiT0clKnByb3RlY3RlZEc2I0YlIiIp

Ky1JInRHNiIiKUBuL1YiIiEiKk92dSlvIiIjISxfcGRJTSIiIiUhLSMqXFImKipRIyIiJy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIik=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNz

op(2,e6);

KiQpSSJ0RzYiIiRXIiIiIg==

op(6,e6);

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

Second relation between S1 and S2

eqS1S22:=op(1,op(6,e6));

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

Finally we eliminate S2 to get an equation for S1

e7:=factor(resultant(eqS1S21,eqS1S22,S2));

LCQqKiIkYyMiIiIpSSJ0RzYiIiM9RiUsUiooIiYkbz5GJSlJI1MxR0YoIiIpRiUpRiciIihGJUYlKigiJksoeUYlKUYuRjFGJSlGJyIiJ0YlISIiKigiJiUpKnBGJSlGLkY2RiVGMEYlRjcqKCInMk04RiVGOkYlKUYnIiImRiVGJSooIidfKjQjRiUpRi5GPkYlRjVGJUYlKigiJidwTEYlKUYuIiIlRiVGMEYlRjcqKCInZlk3RiVGQUYlKUYnRkVGJUY3KigiJyNcUCNGJUZERiVGPUYlRjcqKCImI1JuRiUpRi4iIiRGJUY1RiVGJSooIiU/XkYlKUYuIiIjRiVGMEYlRjcqKCImeiZwRiVGREYlKUYnRk5GJUYlKigiJ2tdN0YlRk1GJUZIRiVGJSooIiY7JWVGJUZRRiVGPUYlRjcqKEZQRiVGLkYlRjVGJUYlKiZGJEYlRjBGJUY3KigiJlpLI0YlRk1GJSlGJ0ZSRiVGNyooIiYhb0dGJUZRRiVGVUYlRjcqKCImP1ojRiVGLkYlRkhGJUYlKiYiJTs3RiVGPUYlRjcqKCIlN1ZGJUZRRiVGJ0YlRiUqKCIlUzZGJUYuRiVGaG5GJUYlKiYiJSEzJUYlRlVGJUY3KiYiJFYkRiVGLkYlRjcqJkZmb0YlRidGJUYlRiUpLDwqKCInczU4RiVGUUYlKUYnRi9GJUYlKigiJndYI0YlRlFGJUY1RiVGJSooRltwRiVGLkYlRjBGJUY3KiYiJk9iJ0YlRlxwRiVGJSooIiVPOkYlRlFGJUZIRiVGJSooRl5wRiVGLkYlRj1GJUY3KiYiJiVRO0YlRjVGJUYlKigiI0tGJUZRRiVGaG5GJUYlKihGY3BGJUYuRiVGVUYlRjcqJiIlIW8oRiVGSEYlRiUqKEZocEYlRi5GJUYnRiVGNyomIiQ/JEYlRmhuRiVGJSIiKkYlRlJGJUYl

n:=7: for i from 3 to nops(e7) do normal(series(subs(S1=coeff(Rser(n),x,0),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(i,e7)),t,n+1));od;

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

Ky1JInRHNiIiIyIpIiIhIiUlPSYiIiMiJzM9PiIiJSIoKytDJCIiJy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIik=

algS1:=collect(op(3,e7),S1,factor);

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

And now we eliminate S1 to get an equation for S2

e8:=factor(resultant(eqS1S21,eqS1S22,S1));

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

n:=7: for i from 3 to nops(e8) do normal(series(subs(R0=coeff(Rser(n),x,0),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(i,e8)),t,n+1));od;

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Ky1JInRHNiIiJ1w7ViIiISIpJ1wjRycpIiIjIitvKDNFYCoiIiUiLSdwPHpyJXAiIictSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

algS2:=collect(op(3,e8),S2,factor);

LDQqKCInVDlgIiIiKUkjUzJHNiIiIilGJSlJInRHRigiIzlGJUYlKioiJyVIYSRGJSlGKyIjN0YlLCYqJkYpRiUpRisiIiNGJUYlIiImRiVGJSlGJyIiKEYlISIiKioiJiRvPkYlKUYrIiM1RiUsKComIiRzI0YlKUYrIiIlRiVGJSomIiQ7JEYlRjNGJUYlIiREIkYlRiUpRiciIidGJUYlKioiJEgoRiUpRitGKUYlLCoqJiIlM2lGJSlGK0ZGRiVGJSomIiYlUTpGJUZARiVGJSomIiUzXEYlRjNGJUYlIiVqREYlRiUpRidGNUYlRjgqKiIkViNGJUZNRiUsLComIiUhWylGJUZJRiVGJSomIiYrRyVGJUZNRiVGJSomIiYvMSNGJUZARiVGJSomIiU7SUYlRjNGJUY4IiV0TUYlRiUpRidGQUYlRiUqKiIjRkYlRkBGJSwuKiYiJkMtI0YlRjtGJUYlKiYiJ29mREYlRklGJUYlKiYiJm8mXEYlRk1GJUY4KiYiJlMvI0YlRkBGJUYlKiYiJiNHX0YlRjNGJUY4IiVQJSlGJUYlKUYnIiIkRiVGOCoqRjRGJUYzRiUsMComIiZvQiVGJUYvRiVGJSomIigjZnU3RiVGO0YlRiUqJiIoIW81PUYlRklGJUY4KiYiJ3dUISlGJUZNRiVGOComIidZPElGJUZARiVGJSomIidEJFIjRiVGM0YlRjgiJnBxIkYlRiUpRidGNEYlRiUqJiwyKiYiJW9yRiVGKkYlRjgqJiInO3dXRiVGL0YlRjgqJiIoMz02I0YlRjtGJUYlKiYiJ1d0ISlGJUZJRiVGOComIidPaEFGJUZNRiVGOComIidDInkjRiVGQEYlRjgqJiImaT4mRiVGM0YlRiUiJSg+I0Y4RiVGJ0YlRiUqJkYzRiUsMComIiRjI0YlRi9GJUYlKiYiJlMlSEYlRjtGJUYlKiYiJzdURkYlRklGJUYlKiYiJk9qJUYlRk1GJUY4KiYiJl9GJUYlRkBGJUY4KiYiJiMqKkhGJUYzRiVGOEZbckYlRiVGJQ==

The equations for R0 and S2, both of degree 8

algS1:=19683*S1^8*t^7-78732*S1^7*t^6-2187*t^5*(32*t^2-61)*S1^6+6561*t^4*(32*t^2-19)*S1^5-81*t^3*(416*t^4+2932*t^2-859)*S1^4+81*t^2*(832*t^4+1544*t^2-287)*S1^3-8*t*(640*t^6+7302*t^4+3585*t^2-539)*S1^2+(5120*t^6+24720*t^4+1140*t^2-343)*S1-t*(256*t^6+1216*t^4+4080*t^2-343); algS2:=531441*S2^8*t^14-354294*t^12*(8*t^2+5)*S2^7+19683*t^10*(272*t^4+316*t^2+125)*S2^6-729*t^8*(6208*t^6+15384*t^4+4908*t^2+2563)*S2^5+243*t^6*(8480*t^8+42800*t^6+20604*t^4-3016*t^2+3473)*S2^4-27*t^4*(20224*t^10+255968*t^8-49568*t^6+20440*t^4-52282*t^2+8437)*S2^3+2*t^2*(42368*t^12+1274592*t^10-1810680*t^8-804176*t^6+301746*t^4-239325*t^2+17069)*S2^2+(-7168*t^14-447616*t^12+2111808*t^10-807344*t^8-226136*t^6-278124*t^4+51962*t^2-2197)*S2+t^2*(256*t^12+29440*t^10+274112*t^8-46336*t^6-42752*t^4-29992*t^2+2197);

<Text-field style="Heading 2" layout="Heading 2">Parametrization for S1 and S2 (for M)</Text-field>

We note that S1=R(0) is odd in t

subs(x=0,Rser(9));

LCwqJiIlJ0ckIiIiKUkidEc2IiIiKkYlRiUqJiIkLCRGJSlGJyIiKEYlRiUqJiIjSkYlKUYnIiImRiVGJSomIiIlRiUpRiciIiRGJUYlRidGJQ==

We use the parametrization given by van Hoeij for Gessel walks (Tm, m like 'Mark')

t2Tm:=(Tm-1)/256/Tm^3*(Tm+3)^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(5);Tmser(6);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

res:=numer(factor(subs(S1=t*S1st,t=sqrt(t2Tm),algS1))):nops(res);

IiIk

n:=6:normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(2,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(3,res)),t,n+1));

KydJInRHNiIsJComIiM7IiIiLUklY3Nnbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiNGKEYoRigtSSJPR0YsNiNGKCIiJg==

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KydJInRHNiIiMXMnM0wmKkgmMzoiIiEtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIj

This is still quadratic over Q(Tm)

algS1stTm:=collect(op(2,res),S1st,factor);

LCgqKCksJkkjVG1HNiIiIiIiIiRGKCIiJ0YoLCZGJkYoRighIiJGKClJJVMxc3RHRiciIiNGKEYoKioiJGMjRigpRiZGKUYoKUYlRilGKEYuRihGLCooIiVDNUYoRjJGKCwqKiRGMkYoRigqJiIjNkYoKUYmRi9GKEYsKiYiIiZGKEYmRihGLEYoRixGKEYs

factor(discrim(algS1stTm,S1st));

LCQqKiIlJzQlIiIiKSwmSSNUbUc2IkYlRiVGJSIiJUYlKSwmRihGJSIiJEYlIiInRiUpRihGLUYlRiU=

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

res:=factor(subs(Tm=Zm^2,algS1stTm));

KiYsOiomSSVTMXN0RzYiIiIiKUkjWm1HRiYiIihGJ0YnKiZGJUYnKUYpIiInRidGJyooIiIqRidGJUYnKUYpIiImRidGJyomIiNLRidGLEYnRicqKEYvRidGJUYnKUYpIiIlRidGJyomIiMnKkYnRjBGJyEiIiooIiNGRidGJUYnKUYpIiIkRidGJyomRjNGJ0Y1RidGOSooRjtGJ0YlRicpRikiIiNGJ0YnKiZGM0YnRjxGJ0Y5KihGO0YnRiVGJ0YpRidGJyomRjtGJ0YlRidGJ0YnLDpGJEYnRitGOUYuRidGMkY5RjRGOUY3RjlGOkYnRj5GJ0Y/RjlGQkY5RkNGJ0ZERjlGJw==

n:=6:normal(series(subs(Zm=Zmser(n),S1st=coeff(Rser(n),x,0)/t,op(1,res)),t,n+1)); normal(series(subs(Zm=Zmser(n),S1=coeff(Rser(n),x,0)/t,op(1,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIsJiomIiRHIiIiIkklUzFzdEdGJEYoRihGJyEiIiIiISwmKiYiJDcmRihGKUYoRigiJUM1RioiIiMsJiomIiVbX0YoRilGKEYoIiZrNyJGKiIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIiJw==

S1stZm:=factor(solve(op(1,res),S1st));

LCQqLCIjSyIiIilJI1ptRzYiIiIkRiUsKiokRiZGJUYlKiZGKUYlKUYnIiIjRiUhIiJGJ0YvRiVGL0YlLCZGJ0YlRiVGJUYvKSwmKiRGLUYlRiVGKUYlRilGL0Yv

Now for S2, which is even

coeff(Sser(8),x,2);

LCoqJiIlNTciIiIpSSJ0RzYiIiIpRiVGJSomIiQxIkYlKUYnIiInRiVGJSomIiM1RiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

res:=numer(factor(subs(t=sqrt(t2Tm),algS2))): nops(res);

IiIj

n:=6:normal(series(subs(Tm=Tmser(n),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiMSlbJipbPTNEbSoiIiEiMytPZjwkKXkxdlgiIiMiNXdobS5KZydvclAiIiIlLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiJw==

algS2Tm:=collect(op(1,res),S2,factor);

LCgqKCksJkkjVG1HNiIiIiIiIiRGKCIiJ0YoLCZGJkYoRighIiJGKClJI1MyR0YnIiIjRihGKCoqIiIlRigsLCokKUYmRjFGKEYoKiYiI1tGKClGJkYpRihGKComRipGKClGJkYvRihGLComIiNDRihGJkYoRihGKUYsRigpRiVGKUYoRi5GKEYsKihGMUYoRitGKCwwKiQpRiZGKkYoRigqJiIkZSJGKClGJiIiJkYoRiwqJiIkXipGKEY0RihGKComRjFGKEY3RihGLComIiReJEYoRjlGKEYoKiYiJEUiRihGJkYoRiwiIipGKEYoRig=

res:=factor(subs(Tm=Zm^2,algS2Tm));

KiYsQiomSSNTMkc2IiIiIilJI1ptR0YmIiIoRidGJyomRiVGJylGKSIiJ0YnRicqJiIiI0YnRihGJyEiIiooIiIqRidGJUYnKUYpIiImRidGJyomIiNJRidGLEYnRicqKEYyRidGJUYnKUYpIiIlRidGJyomIiNtRidGM0YnRjAqKCIjRkYnRiVGJylGKSIiJEYnRicqJkYvRidGOEYnRjAqKEY9RidGJUYnKUYpRi9GJ0YnKiYiIzVGJ0Y+RidGJyooRj1GJ0YlRidGKUYnRicqJiIjVUYnRkJGJ0YnKiZGPUYnRiVGJ0YnKiZGLUYnRilGJ0YwRi1GMEYnLEJGJEYnRitGMEYuRjBGMUYnRjVGMEY3RjBGOkYwRjxGJ0ZARidGQUYwRkNGJ0ZFRidGRkYwRkhGMEZJRjBGLUYnRic=

n:=6:normal(series(subs(Zm=Zmser(n),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(1,res)),t,n+1)); normal(series(subs(Zm=Zmser(n),S2=1/2*subs(x=0,diff(Sser(n),x$2)),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIhJEciIiIhISVfNiIiIyEmJ3A4IiIlLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiJw==

S2Zm:=factor(solve(op(1,res),S2));

LCQqLCIiIyIiIiwmSSNabUc2IkYlRiUhIiJGJSwwKiQpRiciIidGJUYlKiYiIzlGJSlGJyIiJkYlRikqJiIjPkYlKUYnIiIlRiVGJSomIiM/RiUpRiciIiRGJUYlKiYiIzpGJSlGJ0YkRiVGJSomRi1GJUYnRiVGKUY5RilGJSwmRidGJUYlRiVGKSksJiokRjxGJUYlRjlGJUY5RilGJQ==

Results of this subsection

t2Tm:=(1/256)*(Tm-1)*(Tm+3)^3/Tm^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(5);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

algTm:=numer(t^2-t2Tm); algZm:=subs(Tm=Zm^2,algTm);

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

S1stZm:=-32*Zm^3*(Zm^3-3*Zm^2-Zm-1)/((Zm+1)*(Zm^2+3)^3); S2Zm:=(2*(Zm-1))*(Zm^6-14*Zm^5+19*Zm^4+20*Zm^3+15*Zm^2-6*Zm-3)/((Zm+1)*(Zm^2+3)^3);

LCQqLCIjSyIiIilJI1ptRzYiIiIkRiUsKiokRiZGJUYlKiZGKUYlKUYnIiIjRiUhIiJGJ0YvRiVGL0YlLCZGJ0YlRiVGJUYvKSwmKiRGLUYlRiVGKUYlRilGL0Yv

The value of M00

M00:=S1stZm :

eqS:=-Sx*(Sx+2)*(Sx+1)*(t*x^2+2*t*x+t-x)*(t*x^2-2*t*x+t-x)/x^2+t*(3*S1^2*t*x+2*S2*t*x-5*S1*x+x^2+1)*Sx/x-2*t*(S1*t-1)*x+t*(3*S1^2*t+2*S2*t-5*S1)+2*t^2*S1/x;

<Text-field style="Heading 2" layout="Heading 2">Back to S(x); parametrization (for M)</Text-field>

Sser(4);

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An equation between S(x), t, x and Zm

algSZmt:=collect(numer(factor(subs(S1=t*S1stZm,S2=S2Zm,eqS))),Sx,factor): nops(factor(%));

IiRPJA==

We can decrease the degree in t using the expression of t^2 in terms of Tm

t2Zm:=subs(Tm=Zm^2,t2Tm);

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algSZmsmall:=collect(op(4,factor(rem(algSZmt,numer(t^2-t2Zm),t))),[Sx,Zm],factor);

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

n:=10: normal(series(subs(Zm=Zmser(n),Sx=Sser(n),algSZmsmall),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Let us check that S(x) has degree 24 over Q(x,t)

res:=numer(factor(resultant(algSZmsmall,algZm,Zm))): nops(%);

IiVuTw==

degree(res,Sx);

IiND

S(xt) is even in t

n:=10: series(subs(x=x*t,Sser(n+1)),t,n+1);

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res:=numer(factor(subs(x=x*t,Sx=Sxt,t=sqrt(t2Zm),algSZmsmall))): nops(res);

IiIl

op(1,res); op(2,res);op(3,res);

LCZJI1ptRzYiIiIiRiUhIiI=

LCZJI1ptRzYiIiIiRiVGJQ==

KiQpLCYqJClJI1ptRzYiIiIjIiIiRioiIiRGKkYrRio=

nops(op(4,res));

IiN1

This is the equation satisfied by S(xt) over Q(x,Zm)

algSxtZm:=collect(op(4,res),Sxt,factor);

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Parametrisation of x

xTmUm:=factor(solve(16*Tm^2*(Um^2-Tm)-x*(-2*Tm+Um+Um*Tm)*(-9*Tm+Um^2+8*Tm*Um+Tm^2-Tm*Um^2),x));

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xZmUm:=factor(subs(Tm=Zm^2,xTmUm));

LCQqMCIjOyIiIilJI1ptRzYiIiIlRiUsJkkjVW1HRihGJUYnISIiRiUsJkYrRiVGJ0YlRiUsKComRitGJSlGJyIiI0YlRiUqJkYxRiVGMEYlRixGK0YlRiwsKiomRitGJUYnRiVGJSokRjBGJUYlRitGJSomIiIkRiVGJ0YlRixGLCwqRjRGJUY1RixGK0YsRjZGLEYsRiw=

Expansion of Um in terms of (Tm-1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

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Let us check

n:=9:normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),numer(x-xTmUm)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

res:=numer(factor(subs(x=xTmUm,Tm=Zm^2,algSxtZm))):nops(%);indets(%%);

IiIl

PCVJJFN4dEc2IkkjVW1HRiRJI1ptR0Yk

op(1,res);op(2,res);

IiZPYic=

KiQpSSNabUc2IiIjNyIiIg==

degree(op(3,res),Sxt);degree(op(4,res),Sxt);

IiIi

IiIj

the quadratic term does not vanish

n:=10: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Zm^2, Zm=Zmser(n),Sxt=subs(x=x*t,Sser(n)),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=20: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Zm^2,Zm=Zmser(n),Sxt=subs(x=x*t,Sser(n)),op(4,res)),t,n+1));

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SxtZmUm:=factor(solve(op(3,res),Sxt));

LCQqKiwmSSNVbUc2IiIiIkkjWm1HRiYhIiJGJywqKiZGJUYnRihGJ0YnKiQpRigiIiNGJ0YnRiVGJyomIiIkRidGKEYnRilGJyw2KiYpRiVGLkYnKUYoRjBGJ0YnKigiIiVGJ0YlRicpRihGNkYnRicqJClGKCIiJkYnRikqJkYzRidGLUYnRikqJiIjNkYnRjdGJ0YpKiZGM0YnRihGJ0YnKihGNkYnRiVGJ0YtRidGJyomIiIqRidGNEYnRikqJEYzRidGKSomRjBGJ0YtRidGKUYnLDYqJilGJUY2RidGN0YnRicqKEYuRidGM0YnKUYoIiInRidGKSokKUYoIiIpRidGJyooIiM7RidGJUYnRkhGJ0YnKigiI0NGJ0YzRidGN0YnRikqJkZQRidGSEYnRikqKCIjW0YnRiVGJ0Y3RidGJyokRkZGJ0YpKihGSUYnRjNGJ0YtRidGKSomRkFGJ0Y3RidGKUYpRik=

Let's check

n:=18: normal(series(subs(Um=convert(UmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),Zm=Zmser(n),SxtZmUm),t,n+1));

KzFJInRHNiJJInhHRiQiIiMsJiokKUYlRiYiIiJGKiomIiIlRipGJUYqRipGLCwoKiQpRiUiIiRGKkYqKiYiIzVGKkYpRipGKiomIiNKRipGJUYqRioiIicsKiokKUYlRixGKkYqKiYiIz1GKkYvRipGKiomIiQxIkYqRilGKkYqKiYiJCwkRipGJUYqRioiIiksLCokKUYlIiImRipGKiomIiNHRipGOEYqRioqJiIkXyNGKkYvRipGKiomIiU1N0YqRilGKkYqKiYiJSdHJEYqRiVGKkYqRjIsLiokKUYlRjVGKkYqKiYiI1NGKkZCRipGKiomIiQtJkYqRjhGKkYqKiYiJSlRJEYqRi9GKkYqKiYiJiplOUYqRilGKkYqKiYiJmkmUUYqRiVGKkYqIiM3LUkiT0clKnByb3RlY3RlZEc2I0YqIiM5

normal(series(subs(x=x*t,Sser(10)),t,10));

Ky1JInRHNiJJInhHRiQiIiMsJiokKUYlRiYiIiJGKiomIiIlRipGJUYqRipGLCwoKiQpRiUiIiRGKkYqKiYiIzVGKkYpRipGKiomIiNKRipGJUYqRioiIicsKiokKUYlRixGKkYqKiYiIz1GKkYvRipGKiomIiQxIkYqRilGKkYqKiYiJCwkRipGJUYqRioiIiktSSJPRyUqcHJvdGVjdGVkRzYjRipGMg==

Baxk to M(0,x)

xZmUm;

op(Sser);

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJXN1YnNHRik2JS9JInhHRiUiIiEvSSJ5R0YlRjAqKC1JJU1zZXJHRiVGIyIiIkYzRjdJInRHRiVGN0Y4RiVGJUYl

M0xtZmUm:=factor(SxtZmUm/xZmUm/t2Zm);

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

let us check this

n:=18: normal(series(subs(Um=convert(UmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),Zm=Zmser(n),M0xtZmUm),t,n+1));

KzFJInRHNiIiIiIiIiEsJiIiJUYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIzVGJUYpRiVGJSIjSkYlRigsKiokKUYpIiIkRiVGJSomIiM9RiVGLUYlRiUqJiIkMSJGJUYpRiVGJSIkLCRGJSIiJywsKiQpRilGKEYlRiUqJiIjR0YlRjNGJUYlKiYiJF8jRiVGLUYlRiUqJiIlNTdGJUYpRiVGJSIlJ0ckRiUiIiksLiokKUYpIiImRiVGJSomIiNTRiVGPUYlRiUqJiIkLSZGJUYzRiVGJSomIiUpUSRGJUYtRiVGJSomIiYqZTlGJUYpRiVGJSImaSZRRiVGLy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

normal(series(subs(x=0,y=x*t,Mser(10)),t,10));

Ky9JInRHNiIiIiIiIiEsJiIiJUYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIzVGJUYpRiVGJSIjSkYlRigsKiokKUYpIiIkRiVGJSomIiM9RiVGLUYlRiUqJiIkMSJGJUYpRiVGJSIkLCRGJSIiJywsKiQpRilGKEYlRiUqJiIjR0YlRjNGJUYlKiYiJF8jRiVGLUYlRiUqJiIlNTdGJUYpRiVGJSIlJ0ckRiUiIiktSSJPRyUqcHJvdGVjdGVkRzYjRiVGLw==

series(%%-%,t,10);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

What to remember from the above section

xTmUm:=-16*Tm^2*(-Um^2+Tm)/((Tm*Um-2*Tm+Um)*(-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm)); xZmUm:=-16*Zm^4*(Um-Zm)*(Um+Zm)/((Um*Zm^2-2*Zm^2+Um)*(Um*Zm+Zm^2+Um-3*Zm)*(Um*Zm-Zm^2-Um-3*Zm));

LCQqLCIjOyIiIilJI1RtRzYiIiIjRiUsJiokKUkjVW1HRihGKUYlISIiRidGJUYlLCgqJkYnRiVGLUYlRiUqJkYpRiVGJ0YlRi5GLUYlRi4sLComRidGJUYsRiVGLiokRiZGJUYlKigiIilGJUYnRiVGLUYlRiVGK0YlKiYiIipGJUYnRiVGLkYuRi4=

Expansion of Um in terms of (Tm-1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

KzNJJFRtMUc2IiIiIiIiISNGJSIiI0YlIyEiIiIiKUYoLCYqJiNGJSIjS0YlSSJ4R0YkRiVGJSNGJSIjO0YlIiIkLCgqJiMiIigiJEciRiVGMEYlRiojIiImRjhGKiomI0YlRjhGJSlGMEYoRiVGJSIiJSwqKiYjIiNQIiQ3JkYlRjBGJUYlI0Y3IiRjI0YlKiYjIiM2RkNGJUY9RiVGKiomI0YlRkNGJSlGMEYzRiVGJUY6LCwqJiNGJSIlWz9GJSlGMEY+RiVGJSMiI0AiJUM1RioqJiNGN0ZTRiVGS0YlRioqJiMiIyIpRk9GJUY9RiVGJSomI0ZIRjhGJUYwRiVGKiIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY3

SxtZmUm :=-(Um-Zm)*(Um*Zm+Zm^2+Um-3*Zm)*(Um^2*Zm^3+4*Um*Zm^4-Zm^5-Um^2*Zm^2-11*Zm^4+Um^2*Zm+4*Um*Zm^2-9*Zm^3-Um^2-3*Zm^2)/(Um^4*Zm^4-2*Um^2*Zm^6+Zm^8+16*Um*Zm^6-24*Um^2*Zm^4-24*Zm^6+48*Um*Zm^4-Um^4-6*Um^2*Zm^2-9*Zm^4);

M0xtZmUm:=16*(Um*Zm+Zm^2+Um-3*Zm)^2*(Um^2*Zm^3+4*Um*Zm^4-Zm^5-Um^2*Zm^2-11*Zm^4+Um^2*Zm+4*Um*Zm^2-9*Zm^3-Um^2-3*Zm^2)*Zm^2*(Um*Zm^2-2*Zm^2+Um)*(Um*Zm-Zm^2-Um-3*Zm)/((Um^4*Zm^4-2*Um^2*Zm^6+Zm^8+16*Um*Zm^6-24*Um^2*Zm^4-24*Zm^6+48*Um*Zm^4-Um^4-6*Um^2*Zm^2-9*Zm^4)*(Um+Zm)*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

algSZmsmall:=((x-1)^2*(x+1)^2*Zm^8+(-512*t*x^3+8*x^4-512*t*x+240*x^2+8)*Zm^6+18*(x-1)^2*(x+1)^2*Zm^4-27*(x-1)^2*(x+1)^2)*Sx^3+(3*(x-1)^2*(x+1)^2*Zm^8+(-1536*t*x^3+24*x^4-1536*t*x+720*x^2+24)*Zm^6+54*(x-1)^2*(x+1)^2*Zm^4-81*(x-1)^2*(x+1)^2)*Sx^2+((2*x^4-20*x^2+2)*Zm^8+(-1280*t*x^3+16*x^4-1280*t*x+688*x^2+16)*Zm^6+(12*(3*x^2-1))*(x^2-3)*Zm^4-144*Zm^2*x^2-54*(x-1)^2*(x+1)^2)*Sx-16*Zm^8*x^2-64*x*t*(x-1)*(x+1)*Zm^7-16*x*(16*t*x^2+16*t-13*x)*Zm^6-128*x*t*(x-1)*(x+1)*Zm^5-48*Zm^4*x^2-64*x*t*(x-1)*(x+1)*Zm^3-144*Zm^2*x^2;

<Text-field style="Heading 2" layout="Heading 2">Determination of M(x,0) (or R(x)) and parametrization</Text-field>

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjAqKEYsRipGKUYqRjZGKkYwRipGKiomRjZGKkYpRjBGKg==

We first find an equation for DD:=S(x)-2S(1/x)-1

indets(algSZmsmall);

PCZJI1N4RzYiSSNabUdGJEkidEdGJEkieEdGJA==

res:=numer(factor(resultant(subs(Sx=DD+2*S1x+1,algSZmsmall),subs(Sx=S1x,x=1/x,algSZmsmall),S1x))): nops(res);

IiIl

seq(nops(op(i,res)),i=1..nops(res));

NiYiIiIiIiMiI04iJClv

seq(degree(op(i,res),DD),i=1..nops(res));

NiYiIiFGIyIiJCIiJw==

Choosing the correct factor

n:=1: normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))-1,Zm=Zmser(n),op(3,res)),t,n+1));

KydJInRHNiIsJComIiVbPyIiIilJInhHRiQiIiNGKEYoIiIhLUkiT0clKnByb3RlY3RlZEc2I0YoRig=

n:=10: normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))-1,Zm=Zmser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

We decrease the degree in t

degree(op(4,res),t);

IiIk

res1:=factor(rem(op(4,res),numer(t^2-t2Zm),t)):nops(%);

IiRZJg==

algDDZm:=res1:

Hence DD has degree 6 over Q(x,Z,t)

degree(algDDZm,DD);

IiIn

But also over Q(x,T,t)

seq(coeff(algDDZm,Zm,2*i+1),i=0..10);

Ni0iIiFGI0YjRiNGI0YjRiNGI0YjRiNGIw==

algDDTm:=collect(subs(Zm=sqrt(Tm),algDDZm),DD,factor): degree(%,t);

IiIi

DD is in fact bicubic

seq(coeff(algDDTm,DD,2*i+1),i=0..2);

NiUiIiFGI0Yj

DD has degree 24 over Q(x,t)

res:=factor(resultant(algDDTm,numer(t^2-t2Tm),Tm)):nops(res);

IiVOVA==

degree(res,DD);

IiND

Nox we return to R(x)

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYwRiopSSNEZUdGJyNGKkYsRipGKiooIiIkRipJInRHRidGKi1JIlJHRidGKEYqRjAqKEYsRipGKUYqRjZGKkYwRipGKiomRjZGKkYpRjBGKg==

res:=numer(factor(subs(DD=(3*t*Rx+2*t*x-1-t/x)/sqrt(Delta),algDDTm))): nops(%);

IiIj

op(1,res);nops(op(2,res));

KiQpSSJ4RzYiIiIjIiIi

IiUtTw==

degree(op(2,res),Rx);degree(op(2,res),t);

IiIn

IiIo

A lower degree in t

res:=numer(factor(rem(op(2,res),numer(t^2-t2Tm),t))): nops(res);

IiIk

op(1,res);op(2,res);

IiNG

LCZJI1RtRzYiIiIiRiUhIiI=

nops(op(3,res));

IiVGRw==

Equation of R(x) over Q(t,x,T), of degree 6

algRTm:=collect(op(3,res),Rx,factor):

degree(algRTm,Rx);degree(algRTm,t);

IiIn

IiIi

Equation for R(x) over Q(x,t): degree 12 only

res:=numer(factor(resultant(algRTm,numer(t^2-t2Tm),Tm))): nops(%);

IiIm

seq(nops(op(i,res)),i=1..nops(res));seq(degree(op(i,res),Rx),i=1..nops(res));

NiciIiIiIiNGJCIkTSQiJGEl

NiciIiFGI0YjIiM3RiQ=

Choosing the correct factor

n:=10:normal(series(subs(Rx=Rser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

n:=3:normal(series(subs(Rx=Rser(n),op(5,res)),t,n+1));

Ky1JInRHNiIsJComIiIlIiIiKUkieEdGJCIjN0YoISIiIiIhLCYqJiIjYUYoKUYqIiM4RihGKComRjBGKClGKiIjNkYoRixGKCwoKiYiJEkkRigpRioiIzlGKEYsKiYiJCMpKUYoRilGKEYoKiYiJHcjRigpRioiIzVGKEYsIiIjLCoqJiIlKzdGKClGKiIjOkYoRigqJiIlJVEnRihGMUYoRiwqJiIlR2ZGKEY0RihGKComIiRHJkYoKUYqIiIqRihGLCIiJC1JIk9HJSpwcm90ZWN0ZWRHNiNGKEYn

algR:=collect(op(4,res),Rx,factor):

degree(algR,Rx);

IiM3

Parity for R(xt)/t

subs(x=x*t,Rser(3));

LCgqJiwmKiYpSSJ0RzYiIiIjIiIiKUkieEdGKEYpRipGKiIiJUYqRiopRiciIiRGKkYqKiZGLkYqRixGKkYqRidGKg==

res:=factor(subs(t=sqrt(t2Tm),factor(subs(x=x*t,Rx=Rxtst*t,algRTm)))):indets(%);

PCVJJlJ4dHN0RzYiSSNUbUdGJEkieEdGJA==

seq(indets(op(i,res)),i=1..nops(res));

Nik8IjwjSSNUbUc2IkYkPCRGJUkieEdGJjwlSSZSeHRzdEdGJkYlRihGKUYk

n:=3: normal(series(subs(Tm=Tmser(n),Rxtst=subs(x=x*t,Rser(n)/t),op(5,res)),t,n+1));

KydJInRHNiIsKiomIilrKTNyJyIiIilJInhHRiQiIiRGKEYoKiYiKidIajE1RigpRioiIiNGKCEiIiomRi1GKEYqRihGMEYnRigiIiEtSSJPRyUqcHJvdGVjdGVkRzYjRihGLw==

n:=10: normal(series(subs(Tm=Tmser(n),Rxtst=subs(x=x*t,Rser(n)/t),op(6,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

R(xt)/t is cubic over Q(x,T)

algRxtstTm:=collect(op(6,res),Rxtst,factor);

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

There we find the correct degree

degree(algRxtstTm,Rxtst);

IiIk

And finally we parametrize x by Um

res:=numer(factor(subs(x=xTmUm,algRxtstTm))): nops(res);

IiIl

seq(indets(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rxtst),i=1..nops(res));

NiY8IjwjSSNUbUc2IjwlSSZSeHRzdEdGJkYlSSNVbUdGJkYn

NiYiIiFGIyIiIiIiIw==

We determine which term vanishes

n:=10: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Rxtst=1/t*subs(x=x*t,Rser(n)),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Rxtst=1/t*subs(x=x*t,Rser(n)),op(4,res)),t,n+1));

KydJInRHNiIhKTtzeDsiIzktSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

Rational expression of R(xt)/t in terms of Tm and Um

RxtstTmUm:=factor(solve(op(3,res),Rxtst));

LCQqMCIjOyIiIkkjVG1HNiJGJSwoKiZGJkYlSSNVbUdGJ0YlRiUqJiIiI0YlRiZGJSEiIkYqRiVGJSwyKiYpRiZGLEYlKUYqRixGJUYtKiQpRiYiIiRGJUYlKihGJEYlRjBGJUYqRiVGJSooRixGJUYmRiVGMUYlRiUqJiIjXUYlRjBGJUYtKihGJEYlRiZGJUYqRiVGLSokRjFGJUYtKiYiIzpGJUYmRiVGLUYlLCwqJkYmRiVGMUYlRi0qJEYwRiVGJSooIiIpRiVGJkYlRipGJUYlRjpGJSomIiIqRiVGJkYlRi1GLSwmRiZGJUYlRi1GLSksJkYmRiVGNEYlRjRGLUYt

Back to M(xt,0)

op(Rser);

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSomSSJ0R0YlIiIiLUklc3Vic0dGKTYkL0kieUdGJSIiIS1JJU1zZXJHRiU2IywmRiRGLkYuISIiRi5GLUknbm9ybWFsR0YpRiVGJUYl

Mxt0TmUm:=RxtstTmUm;

Let us check

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Mxt0TmUm),t,n+1));

KzFJInRHNiIiIiIiIiEsJiIiJUYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIihGJUYpRiVGJSIjSkYlRigsKiokKUYpIiIkRiVGJSomIiM2RiVGLUYlRiUqJiIjbEYlRilGJUYlIiQsJEYlIiInLCwqJClGKUYoRiVGJSomIiM7RiVGM0YlRiUqJiIkRCJGJUYtRiVGJSomIiQjcEYlRilGJUYlIiUnRyRGJSIiKSwuKiQpRikiIiZGJUYlKiYiI0FGJUY9RiVGJSomIiRBI0YlRjNGJUYlKiYiJSZcIkYlRi1GJUYlKiYiJSIqekYlRilGJUYlIiZpJlFGJSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

n:=10: normal(series(subs(x=t*x,y=0,Mser(n)),t,n+1));

series(%%%-%,t,12);

IiIh

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

RxtstTmUm:=-16*Tm*(Tm*Um-2*Tm+Um)*(-Tm^2*Um^2+Tm^3+16*Tm^2*Um+2*Tm*Um^2-50*Tm^2-16*Tm*Um-Um^2-15*Tm)/((-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm)*(Tm-1)*(Tm+3)^3);

Mxt0TmUm:=RxtstTmUm;

And now we repeat the whole procedure with N(x,y) instead of M(x,y)

<Text-field style="Heading 2" layout="Heading 2">Canceling the kernel: an equation between N(x,0), N(0,x) and N(0,1/x)</Text-field>

We cancel the kernel

Paires substituables annulant le noyau : (x,Y), (1/x,Y), (Y,x) et (Y, 1/x)

f1:=collect(subs(Nbis(x,y)=0,y=Y,-eqNbis),[N]); f2:=subs(x=1/x,f1);

LCwiIiIhIiIqKiIiI0YjSSJ0RzYiRiMtSSJOR0YoNiQiIiFGLEYjSSJZR0YoRiRGJCosRidGIywmSSJ4R0YoRiNGI0YkRiMsJkYwRiNGI0YjRiMtRio2JEYsRi1GI0YwRiRGJCoqRiZGI0YnRiMtRio2JEYwRixGI0YtRiRGIyooRidGIy1GKjYkRi1GLEYjRi1GJEYj

LCwiIiIhIiIqKiIiI0YjSSJ0RzYiRiMtSSJOR0YoNiQiIiFGLEYjSSJZR0YoRiRGJCosRidGIywmKiRJInhHRihGJEYjRiNGJEYjLCZGI0YjRjBGI0YjLUYqNiRGLEYtRiNGMUYjRiQqKkYmRiNGJ0YjLUYqNiRGMEYsRiNGLUYkRiMqKEYnRiMtRio2JEYtRixGI0YtRiRGIw==

f3:=collect(subs({x=y,y=x},Nbis(y,x)=0,y=Y,-eqNbis),[N]); f4:=subs(x=1/x,f3);

LCwiIiIhIiIqKiIiI0YjSSJ0RzYiRiMtSSJOR0YoNiQiIiFGLEYjSSJ4R0YoRiRGJCosRidGIywmSSJZR0YoRiNGI0YkRiMsJkYwRiNGI0YjRiMtRio2JEYsRi1GI0YwRiRGJCoqRiZGI0YnRiMtRio2JEYwRixGI0YtRiRGIyooRidGIy1GKjYkRi1GLEYjRi1GJEYj

LCwiIiIhIiIqKiIiI0YjSSJ0RzYiRiMtSSJOR0YoNiQiIiFGLEYjSSJ4R0YoRiNGJCosRidGIywmSSJZR0YoRiNGI0YkRiMsJkYwRiNGI0YjRiMtRio2JEYsKiRGLUYkRiNGMEYkRiQqKkYmRiNGJ0YjLUYqNiRGMEYsRiNGLUYjRiMqKEYnRiMtRio2JEY0RixGI0YtRiNGIw==

indets(f1*f2*f3*f4);

PCtJIllHNiJJInRHRiRJInhHRiQtSSJOR0YkNiQiIiFGIy1GKDYkRipGJi1GKDYkRioqJEYmISIiLUYoNiRGI0YqLUYoNiRGJkYqLUYoNiRGL0Yq

We eliminate the series involving Y

NYsol:=factor(solve({f1,f2},{N(0,Y),N(Y,0)}));

PCQvLUkiTkc2IjYkIiIhSSJZR0YmLCQqLEkieEdGJiIiIiwmLUYlNiQqJEYsISIiRihGLS1GJTYkRixGKEYyRi1GKUYyLCZGLEYtRi1GMkYyLCZGLEYtRi1GLUYyRjIvLUYlNiRGKUYoLCQqJiwqKiZJInRHRiZGLUYvRi1GLSooIiIjRi1GPkYtLUYlNiRGKEYoRi1GMiomRj5GLUYzRi1GLUYpRjJGLUY+RjJGMg==

f3bis:=collect(1/3/t/x/Y*numer(factor(subs(NYsol,f3))),[N],factor);

LCwqKCMiIiMiIiQiIiJJInhHNiIhIiItSSJOR0YpNiQiIiFGLkYnRicqLCNGJ0YmRicsJkkiWUdGKUYnRidGKkYnLCZGMkYnRidGJ0YnRjJGKi1GLDYkRi5GKEYnRioqKEYwRidGKEYqLUYsNiRGKEYuRidGKiooRiRGJ0YoRiotRiw2JCokRihGKkYuRidGKioqRjBGJywmKiZGJUYnRjJGJ0YnRihGKkYnRihGKkkidEdGKUYqRic=

f4bis:=collect(1/3/t/x/Y*numer(factor(subs(NYsol,f4))),[N],factor);

LCwqJiMiIiMiIiQiIiItSSJORzYiNiQiIiFGLEYnRicqLiNGJ0YmRicsJkkiWUdGKkYnRichIiJGJywmRjBGJ0YnRidGJ0YwRjFJInhHRipGMS1GKTYkRiwqJEYzRjFGJ0YxKiZGJEYnLUYpNiRGM0YsRidGMSomRi5GJy1GKTYkRjZGLEYnRjEqKkYuRicsJiooRiVGJ0YwRidGM0YnRidGJ0YxRidGM0YxSSJ0R0YqRjFGJw==

n:=10: normal(series(subs(N(0,x)=subs(x=0,y=x,Nser(n)),N(0,0)=subs(x=0,y=0,Nser(n)),N(0,1/x)=subs(x=0,y=1/x,Nser(n)), N(x,0)=subs(y=0,Nser(n)), N(1/x,0)=subs(x=1/x,y=0,Nser(n)), Y=Y0ser(n+2),f3bis),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

Eliminate N(1/x,0)

f8:=collect(-3*x*f3bis+6*f4bis,[N,x],factor);

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What we should remember from the above subsection

f8 := 2*N(0, 0)+(Y-1)*(Y+1)*x*N(0, x)/Y-(2*(Y-1))*(Y+1)*N(0, 1/x)/(Y*x)-3*N(x, 0)+x/t+2*Y/t-2/(t*x);

LDAqJiIiIyIiIi1JIk5HNiI2JCIiIUYqRiVGJSosLCZJIllHRihGJUYlISIiRiUsJkYtRiVGJUYlRiVJInhHRihGJS1GJzYkRipGMEYlRi1GLkYlKi5GJEYlRixGJUYvRiUtRic2JEYqKiRGMEYuRiVGMEYuRi1GLkYuKiYiIiRGJS1GJzYkRjBGKkYlRi4qJkYwRiVJInRHRihGLkYlKihGJEYlRi1GJUY8Ri5GJSooRiRGJUYwRi5GPEYuRi4=

factor(expand(rationalize(subs(Y=Y0sol,Y-1/Y))));

LCQqKEkieEc2IiEiIkkidEdGJUYmKSomLCoqJkYnIiIiKUYkIiIjRixGLCooRi5GLEYnRixGJEYsRixGJ0YsRiRGJkYsLCpGK0YsRi9GJkYnRixGJEYmRiwjRixGLkYsRiY=

simplify(%/sqrt(Delta),symbolic);

LCQqJEkidEc2IiEiIkYm

Introducing De=Delta

f8form:=1/t*sqrt(De)*(x*N(0,x)-2/x*N(0,1/x))+3*N(x, 0)-2*N(0,0)-x/t-2*Y/t+2/t/x;

LC4qKEkidEc2IiEiIilJI0RlR0YlIyIiIiIiI0YqLCYqJkkieEdGJUYqLUkiTkdGJTYkIiIhRi5GKkYqKihGK0YqRi5GJi1GMDYkRjIqJEYuRiZGKkYmRipGKiomIiIkRiotRjA2JEYuRjJGKkYqKiZGK0YqLUYwNiRGMkYyRipGJiomRi5GKkYkRiZGJiooRitGKkkiWUdGJUYqRiRGJkYmKihGK0YqRi5GJkYkRiZGKg==

n:=10: normal(series(subs(De=Delta,N(0,x)=subs(x=0,y=x,Nser(n)),N(0,0)=subs(x=0,y=0,Nser(n)),N(0,1/x)=subs(x=0,y=1/x,Nser(n)), N(x,0)=subs(y=0,Nser(n)), N(1/x,0)=subs(x=1/x,y=0,Nser(n)), Y=Y0ser(n+2),f8form),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

<Text-field style="Heading 2" layout="Heading 2">An equation between N(0,x) and N(0,1/x) (or S(x) and S(1/x))</Text-field>

Introduce the series R and S

Rser:=proc(n) collect(t*subs(y=0,Nser(n-1)),t,normal): end:

Sser:=proc(n) collect(subs(x=0,y=x,Nser(n)*y*t),t); end:

Rser(6);Sser(4);

LC4qJiwoKiQpSSJ4RzYiIiImIiIiRioqJiIjOUYqKUYnIiIkRipGKiomIiNQRipGJ0YqRipGKilJInRHRigiIidGKkYqKiYsKCokKUYnIiIlRipGKiomIiIqRiopRiciIiNGKkYqIiM2RipGKilGMkYpRipGKiomLCYqJEYtRipGKiomRilGKkYnRipGKkYqKUYyRjhGKkYqKiYsJiokRjtGKkYqRjxGKkYqKUYyRi5GKkYqKiZGJ0YqKUYyRjxGKkYqRjJGKg==

LCwqKEkieEc2IiIiIiwoKiQpRiQiIiVGJkYmKiYiIzdGJilGJCIiI0YmRiYiIzZGJkYmKUkidEdGJSIiJkYmRiYqKEYkRiYsJiokKUYkIiIkRiZGJiomIiInRiZGJEYmRiZGJilGMUYqRiZGJiooRiRGJiwmKiRGLUYmRiZGLkYmRiYpRjFGN0YmRiYqJkYtRiYpRjFGLkYmRiYqJkYkRiZGMUYmRiY=

f13:=collect(subs(N(0,x)=S(x)/t/x,N(0,1/x)=x/t*S(1/x), N(x,0)=R(x)/t,N(0,0)=R(0)/t,Y=(1-t*(x+1/x)-sqrt(De))/2/t,t^2*f8form),[De,R],expand);

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYqRiopSSNEZUdGJyNGKkYsRipGKiooRixGKkkidEdGJ0YqLUkiUkdGJzYjIiIhRipGMCooIiIkRipGNUYqLUY3RihGKkYqRipGMCooRjtGKkY1RipGKUYwRio=

op(1,f13);

KiYsKC1JIlNHNiI2I0kieEdGJiIiIiomIiIjRiktRiU2IyokRighIiJGKUYvRilGKUYpKUkjRGVHRiYjRilGK0Yp

u0:=op(1,f13)^2-subs(De=0,f13)^2;

LCYqJiksKC1JIlNHNiI2I0kieEdGKCIiIiomIiIjRistRic2IyokRiohIiJGK0YxRitGK0YtRitJI0RlR0YoRitGKyokKSwqKihGLUYrSSJ0R0YoRistSSJSR0YoNiMiIiFGK0YxKigiIiRGK0Y3RistRjlGKUYrRitGK0YxKihGPUYrRjdGK0YqRjFGK0YtRitGMQ==

u1:=collect(u0,[S,R],distributed,expand);

LD4qJkkjRGVHNiIiIiIpLUkiU0dGJTYjSSJ4R0YlIiIjRiZGJioqIiIlRiZGJEYmRihGJi1GKTYjKiRGKyEiIkYmRjIqKEYsRiZGJEYmRihGJkYmKihGLkYmRiRGJilGL0YsRiZGJiooRi5GJkYkRiZGL0YmRjIqKEYuRiYpSSJ0R0YlRixGJiktSSJSR0YlNiMiIiFGLEYmRjIqKiIjN0YmRjhGJkY7RiYtRjxGKkYmRiYqJiwmKiZGLkYmRjlGJkYyKihGQEYmRitGMkY4RiZGJkYmRjtGJkYmKigiIipGJkY4RiYpRkFGLEYmRjIqJiwmKiYiIidGJkY5RiZGJiooIiM9RiZGK0YyRjhGJkYyRiZGQUYmRiZGJEYmRiZGMiooRkxGJkY5RiZGK0YyRiYqKEZHRiZGOEYmKUYrRixGMkYy

u1expl:=collect(subs(De=Delta,u0),[S,R],distributed,expand);

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

Extract the negative part in x (which must be zero). Recall that S(0)=0 and D(S)(0)=R(0) by definition of these series

Sser(3);

LCoqKEkieEc2IiIiIiwmKiQpRiQiIiRGJkYmKiYiIidGJkYkRiZGJkYmKUkidEdGJSIiJUYmRiYqKEYkRiYsJiokKUYkIiIjRiZGJkY0RiZGJilGLkYqRiZGJiomRjNGJilGLkY0RiZGJiomRiRGJkYuRiZGJg==

subs(x=0,Rser(n)); coeff(Sser(n),x,1);

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LC4qJiIlLyEpIiIiKUkidEc2IiIjNkYlRiUqJiIkI3lGJSlGJyIiKkYlRiUqJiIjJilGJSlGJyIiKEYlRiUqJkYpRiUpRiciIiZGJUYlKiYiIiNGJSlGJyIiJEYlRiVGJ0Yl

The contribution of the series in x

factor(subs(S(0)=0,R(0)=D(S)(0),series(subs(S(1/x)=0,u1expl),x,2)));

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The series in 1/x

op(4,u1expl)+op(5,u1expl);

LCYqJiwuIiIlIiIiKigiIilGJkkidEc2IkYmSSJ4R0YqISIiRiwqKEYoRiZGK0YmRilGJkYsKiZGKEYmKUYpIiIjRiZGLCooRiVGJilGK0YwRiZGL0YmRiYqKEYlRiZGL0YmRjJGLEYmRiYpLUkiU0dGKjYjKiRGK0YsRjBGJkYmKiYsLkYlRixGJ0YmRi1GJkYuRiZGMUYsRjNGLEYmRjVGJkYm

normal(subs(S(0)=0,series(subs(x=1/x,%),x,3)));

KylJInhHNiIsJCooIiIlIiIiKUkidEdGJCIiI0YoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRighIiJGNiwoKihGJ0YoRilGKClGLEYrRihGKCooRitGKEYpRigtLS1JI0BAR0YvNiRGLkYrRjJGNEYoRjYqKCIiKUYoRipGKEYsRihGKEY1LUkiT0dGMDYjRihGKA==

The negative part

u2:=-4*Sneg(1/x)+convert(%%%,polynom)+%%-subs(x=1/x,convert(%,polynom));

LDQqJiIiJSIiIi1JJVNuZWdHNiI2IyokSSJ4R0YoISIiRiVGLCooIiIpRiUpSSJ0R0YoIiIjRiUpRitGMUYsRiwqKkYkRiVGMEYlLCYqJkYwRiUtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjSSJTR0YoNiMiIiFGJUYlRiVGLEYlRitGLEYsKiYsLkYkRiUqKEYuRiVGMEYlRitGLEYsKihGLkYlRitGJUYwRiVGLComRi5GJUYvRiVGLCooRiRGJUYyRiVGL0YlRiUqKEYkRiVGL0YlRjJGLEYlRiUpLUY9RilGMUYlRiUqJiwuRiRGLEZCRiVGQ0YlRkRGJUZFRixGRkYsRiVGSEYlRiUqKkYkRiVGL0YlRjZGJUYrRiVGJSooRiRGJUYvRiUpRjZGMUYlRiwqKEYxRiVGL0YlLS0tSSNAQEdGOTYkRjhGMUY8Rj5GJUYlKihGLkYlRjBGJUY2RiVGLA==

let us check: this must be non-negative

n:=10: expand(series(subs(De=Delta,Sneg(1/x)=Delta*S(x)*S(1/x),S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,D(S)(0)=subs(x=0,Rser(n)),(D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u2),t,n));

KzVJInRHNiIhIiUiIiMsJComIiIlIiIiSSJ4R0YkRipGKiIiJEYpRiksJComIiM7RipGK0YqRioiIiYsJiomIiIpRiopRitGJkYqRioiIz9GKiIiJywmKiZGM0YqKUYrRixGKkYqKiYiJC8iRipGK0YqRioiIigsKComRjNGKilGK0YpRipGKiomIiMpKUYqRjRGKkYqIiRbIkYqRjMsKComRjNGKilGK0YwRipGKiomIiRDIkYqRjlGKkYqKiYiJGcpRipGK0YqRioiIiotSSJPRyUqcHJvdGVjdGVkRzYjRioiIzU=

and this must be negative

n:=14: expand(series(subs(De=Delta,Sneg(1/x)=0,S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)) ,D(S)(0)=subs(x=0,Rser(n)), (D@@2)(S)(0)=subs(x=0,diff(Sser(n),x$2)), u2),t,n));

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

u2form:=collect((-4*Sneg(1/x)-8*t^2/x^2-4*t*(t*(D(S))(0)-1)/x+4*De*S(1/x)^2-4*De*S(1/x)+4*t^2*(D(S))(0)*x-4*t^2*(D(S))(0)^2+2*t^2*((D@@2)(S))(0)-8*t*(D(S))(0))/4,[De,x],factor);

LDIqKC1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCZGJEYrRitGKkYrSSNEZUdGJkYrRistSSVTbmVnR0YmRidGKiooIiIjRispSSJ0R0YmRjFGKylGKUYxRipGKiooRjNGKywmKiZGM0YrLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlNiMiIiFGK0YrRitGKkYrRilGKkYqKigjRitGMUYrRjJGKy0tLUkjQEBHRjs2JEY6RjFGPkY/RitGKyooRjFGK0YzRitGOEYrRioqKEYyRitGOEYrRilGK0YrKiZGMkYrKUY4RjFGK0Yq

u3form:=S(1/x)*(S(1/x)-1)*De-Sneg(1/x)-F0*x-F1-F2/x-2*t^2/x^2;

LC4qKC1JIlNHNiI2IyokSSJ4R0YmISIiIiIiLCZGJEYrRitGKkYrSSNEZUdGJkYrRistSSVTbmVnR0YmRidGKiomSSNGMEdGJkYrRilGK0YqSSNGMUdGJkYqKiZJI0YyR0YmRitGKUYqRioqKCIiI0YrKUkidEdGJkY2RispRilGNkYqRio=

F0expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form),x,1); F1expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form),x,0); F2expr:=-coeff(subs(Sneg(1/x)=0,De=0,u2form),x,-1);

LCQqJilJInRHNiIiIiMiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSJTR0YmNiMiIiFGKCEiIg==

LCgqKCMiIiIiIiNGJSlJInRHNiJGJkYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYkSSJER0YuRiY2I0kiU0dGKTYjIiIhRiUhIiIqKEYmRiVGKEYlLS1GMkYzRjVGJUYlKiZGJ0YlKUY5RiZGJUYl

KiZJInRHNiIiIiIsJiomRiNGJS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJIlNHRiQ2IyIiIUYlRiVGJSEiIkYl

We now reconstruct Delta S(x) S(1/x)

u5form:=collect((factor( De*S(x)*S(1/x)-solve(u3form,Sneg(1/x))-P0-subs(x=1/x,solve(u3form,Sneg(1/x)) ))),[De,F0,F1,F2],expand);

LDAqJiwsKiQpLUkiU0c2IjYjSSJ4R0YpIiIjIiIiISIiKiZGJ0YtLUYoNiMqJEYrRi5GLUYtKiQpRjBGLEYtRi5GJ0YtRjBGLUYtSSNEZUdGKUYtRi0qJiwmRitGLUYyRi1GLUkjRjBHRilGLUYtKiZGLEYtSSNGMUdGKUYtRi0qJkY3Ri1JI0YyR0YpRi1GLSooRixGLSlGK0YsRi0pSSJ0R0YpRixGLUYtSSNQMEdGKUYuKihGLEYtRj9GLUY+Ri5GLQ==

Let us check

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM6

What to remember from the previous sub-section

Rser:=proc (n) collect(t*subs(y = 0, Nser(n-1)), t, normal) end proc:

Sser:=proc(n) collect(subs(x=0,y=x,Nser(n)*y*t),t) end proc:

f13:=(S(x)-2*S(1/x)+1)*sqrt(De)-2*t*R(0)+3*t*R(x)-1+3*t/x;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYqRiopSSNEZUdGJyNGKkYsRipGKiooRixGKkkidEdGJ0YqLUkiUkdGJzYjIiIhRipGMCooIiIkRipGNUYqLUY3RihGKkYqRipGMCooRjtGKkY1RipGKUYwRio=

u5form:=(-S(x)^2+S(x)*S(1/x)-S(1/x)^2+S(x)+S(1/x))*De+(x+1/x)*F0+2*F1+(x+1/x)*F2+2*x^2*t^2-P0+2*t^2/x^2;

F0expr:=-t^2*(D(S))(0); F1expr:=-(1/2)*t^2*((D@@2)(S))(0)+2*t*(D(S))(0)+t^2*(D(S))(0)^2 ; F2expr:=t*(t*(D(S))(0)-1) ;

LCQqJilJInRHNiIiIiMiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSJTR0YmNiMiIiFGKCEiIg==

LCgqKCMiIiIiIiNGJSlJInRHNiJGJkYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYkSSJER0YuRiY2I0kiU0dGKTYjIiIhRiUhIiIqKEYmRiVGKEYlLS1GMkYzRjVGJUYlKiZGJ0YlKUY5RiZGJUYl

KiZJInRHNiIiIiIsJiomRiNGJS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJIlNHRiQ2IyIiIUYlRiVGJSEiIkYl

<Text-field style="Heading 2" layout="Heading 2">An equation for N(0,x) (or S(x))</Text-field>

The right multiplication

u11:=collect(expand(u5form*(-2+S(x)+S(1/x))),[De,S],distributed,expand);

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

factor(u11/subs(x=1/x,u11));

IiIi

Extract the non-negative part. We first look at the terms other than S(1/x)

normal(subs(S(0)=0,series(subs(S(1/x)=0,De=Delta,u11),x,2)));

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The terms involving S(1/x)

u11-subs(S(1/x)=0,u11);

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normal(subs(S(0)=0,series(subs(x=1/x,De=Delta,%),x,3)));

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eqSp:=subs(S(1/x)=0,u11)-convert(%%%,polynom)+convert(%,polynom);

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Let us check this equation

n:=16: normal(series(subs(S(x)=Sser(n),De=Delta,F0=F0expr, F1=F1expr,F2=F2expr,(D(S))(0)=subs(x=0,Rser(n)),((D@@2)(S))(0)=subs(x=0,diff(Sser(n),x$2)),P0=coeff(Delta*Sser(n)*subs(x=1/x,Sser(n)),x,0), eqSp),t,n+1));

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How to rewrite this ? How many unknown series in t do we really have?

eqSp1:=collect(subs(F1=F1expr,F2=F2expr,F0=F0expr,eqSp),[De,S(x),x],factor);

LDIqJiwoKiQpLUkiU0c2IjYjSSJ4R0YpIiIkIiIiISIiKiZGLEYtKUYnIiIjRi1GLSomRjFGLUYnRi1GLkYtSSNEZUdGKUYtRi0qJiwyKihGMUYtKUYrRjFGLSlJInRHRilGMUYtRi0qJkYrRi1GOUYtRi4qKEYxRi1GOEYtKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGKDYjIiIhRjFGLUYtKiZGOEYtLS0tSSNAQEdGQDYkRj9GMUZDRkRGLUYuKigiIiVGLUY5Ri1GPUYtRi1JI1AwR0YpRi4qJkY5Ri1GK0YuRi4qKEYxRi1GOEYtRjdGLkYtRi1GJ0YtRi0qKEZNRi1GN0YtRjhGLUYuKihGMUYtRitGLUY5Ri1GLSomRjhGLUY8Ri1GLiooRjFGLUY4Ri1GR0YtRi0qKCIiJkYtRjlGLUY9Ri1GLiomRjFGLUZORi1GLQ==

factor(subs(S(0)=0,series(subs(De=Delta,eqSp1),x,3)));

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P0sol:=factor(solve(op(1,%),P0));

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Only one unknown series, namely S1=R0

eqSp2:=collect(subs(P0=P0sol,eqSp1),[De,S(x),x],factor);

LCwqJiwoKiQpLUkiU0c2IjYjSSJ4R0YpIiIkIiIiISIiKiZGLEYtKUYnIiIjRi1GLSomRjFGLUYnRi1GLkYtSSNEZUdGKUYtRi0qJiwsKihGMUYtKUYrRjFGLSlJInRHRilGMUYtRi0qJkYrRi1GOUYtRi4qKkYsRi0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjlGLSwmKiZGOUYtRjxGLUYtRi1GLUYtRi0qJkY5Ri1GK0YuRi4qKEYxRi1GOEYtRjdGLkYtRi1GJ0YtRi0qKCIiJUYtRjdGLUY4Ri1GLiooRjFGLUYrRi1GOUYtRi1GO0Yu

What needs to be remembered from this subsection

eqSp2:=(-S(x)^3+3*S(x)^2-2*S(x))*De+(2*x^2*t^2-x*t+3*(D(S))(0)*t*(t*(D(S))(0)+1)-t/x+2*t^2/x^2)*S(x)-4*x^2*t^2+2*x*t-3*(D(S))(0)*t*(t*(D(S))(0)+1);

LCwqJiwoKiQpLUkiU0c2IjYjSSJ4R0YpIiIkIiIiISIiKiZGLEYtKUYnIiIjRi1GLSomRjFGLUYnRi1GLkYtSSNEZUdGKUYtRi0qJiwsKihGMUYtKUYrRjFGLSlJInRHRilGMUYtRi0qJkY5Ri1GK0YtRi4qKkYsRi0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjlGLSwmKiZGOUYtRjxGLUYtRi1GLUYtRi0qJkY5Ri1GK0YuRi4qKEYxRi1GOEYtRjdGLkYtRi1GJ0YtRi0qKCIiJUYtRjdGLUY4Ri1GLiooRjFGLUY5Ri1GK0YtRi1GO0Yu

n:=10: normal(series(subs(De=Delta,S(x)=Sser(n), D(S)(0)=subs(x=0,Rser(n)), eqSp2),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

<Text-field style="Heading 2" layout="Heading 2">The (generalized) quadratic method: algebraicity (for N)</Text-field>

eqS:=map(factor,subs(De=Delta,S(x)=Sx,D(S)(0)=S1,eqSp2));

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Differentiating with respect to x_0

eqSS:=diff(eqS,Sx);

LCoqLCwmSSNTeEc2IiIiIkYnISIiRicsJkYlRiciIiNGKEYnLCoqJkkidEdGJkYnKUkieEdGJkYqRidGJyooRipGJ0YtRidGL0YnRidGLUYnRi9GKEYnLCpGLEYnRjBGKEYtRidGL0YoRidGLkYoRigqLEYlRidGKUYnRitGJ0YxRidGLkYoRigqLEYlRidGJEYnRitGJ0YxRidGLkYoRigqKEYtRicsLioqIiIkRicpSSNTMUdGJkYqRidGLUYnRi5GJ0YnKihGKkYnRi1GJylGLyIiJUYnRicqKEY3RidGOUYnRi5GJ0YnKiQpRi9GN0YnRigqJkYqRidGLUYnRidGL0YoRidGLkYoRic=

How many roots to eqSS?

op(Sser);

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numer(factor(subs(Sx=t*x*Mx,eqSS)));

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

We thus have one root X to eqSS

subs(t=0,%);

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Let us determine the first coefficients

with(gfun):

n:=20:expr:=normal(series(subs(Sx=Sser(n), S1=subs(x=0,Rser(n)), eqSS),t,n+1)):

Xisol:=algeqtoseries(numer(convert(expr,polynom)),t,x,10,true);

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The x derivative

eqSx:=collect(diff(eqS,x),[Sx,S1],factor);

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We now want to find an equation for S1 by eliminating X and S(X). (An alternative is to say that the discriminant of eqS w.r.t. S has a double root, as a polynomial in x.)

Elimination of Sx. This is the discriminant with respect to Sx

e1:=numer(factor(resultant(eqS,eqSS,Sx))):nops(e1);

IiIk

op(1,e1); op(2,e1);

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KiQpLCoqJkkidEc2IiIiIilJInhHRiciIiNGKEYoKihGK0YoRiZGKEYqRighIiJGJkYoRipGLUYrRig=

e2:=op(3,e1):nops(e2);indets(e2);

IiNr

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and then of x

e3:=numer(factor(resultant(numer(eqSS),numer(eqSx),Sx))):nops(e3);

IiIk

op(1,e3);op(2,e3);

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e4:=op(3,e3): nops(%);indets(e4);

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res:=factor(resultant(e2,e4,x)):nops(res);

IiIo

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

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n:=10: for i from 3 to nops (res) do normal(series(subs(S1=subs(x=0,Rser(n)),op(i,res)),t,n+1));od;

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op(5,res);

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algS1:=collect(op(1,%),S1,factor);

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The equations for S1, of degree 8

algS1:=27*S1^8*t^7+108*S1^7*t^6+189*S1^6*t^5+189*S1^5*t^4-9*t^3*(32*t^4+28*t^2-13)*S1^4-9*t^2*(64*t^4+56*t^2-5)*S1^3-2*t*(256*t^6-312*t^4+156*t^2-5)*S1^2-(32*t^2-1)*(4*t^2-6*t+1)*(4*t^2+6*t+1)*S1-t*(256*t^6+576*t^4-48*t^2+1);

<Text-field style="Heading 2" layout="Heading 2">Parametrization for S1 (series N)</Text-field>

We note that S1 is odd in t

subs(x=0,Rser(9));

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We use the parametrization given by van Hoeij for Gessel walks (Tm, m like 'Mark')

t2Tm:=(Tm-1)/256/Tm^3*(Tm+3)^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(5);Tmser(6);

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KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

res:=numer(factor(subs(S1=t*S1st,t=sqrt(t2Tm),algS1))):nops(res);

IiIk

n:=10:normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(2,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S1st=coeff(Rser(n),x,0)/t,op(3,res)),t,n+1));

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This is quadratic over Q(x,T)

algS1stTm:=collect(op(2,res),S1st,factor);

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factor(discrim(algS1stTm,S1st));

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Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

res:=factor(subs(Tm=Zm^2,algS1stTm));

KiYsOiomSSVTMXN0RzYiIiIiKUkjWm1HRiYiIihGJ0YnKiZGJUYnKUYpIiInRichIiIqKCIiKkYnRiVGJylGKSIiJkYnRicqJiIjS0YnRixGJ0YnKihGMEYnRiVGJylGKSIiJUYnRi4qJiIjJypGJ0YxRidGJyooIiNGRidGJUYnKUYpIiIkRidGJyomRjlGJ0Y2RidGJyooRjtGJ0YlRicpRikiIiNGJ0YuKiZGOUYnRjxGJ0YuKihGO0YnRiVGJ0YpRidGJyomRjtGJ0YlRidGLkYnLDpGJEYnRitGJ0YvRidGM0YuRjVGJ0Y4RidGOkYnRj5GLkY/RidGQkYuRkNGJ0ZERidGJw==

n:=6:normal(series(subs(Zm=Zmser(n),S1st=coeff(Rser(n),x,0)/t,op(1,res)),t,n+1)); normal(series(subs(Zm=Zmser(n),S1st=coeff(Rser(n),x,0)/t,op(2,res)),t,n+1));

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KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

S1stZm:=factor(solve(op(2,res),S1st));

LCQqLCIjSyIiIilJI1ptRzYiIiIkRiUsKiokRiZGJUYlKiZGKUYlKUYnIiIjRiUhIiIqJkYpRiVGJ0YlRiVGKUYlRiUsJkYnRiVGJUYlRi8pLCYqJEYtRiVGJUYpRiVGKUYvRiU=

Results of this subsection

t2Tm:=(1/256)*(Tm-1)*(Tm+3)^3/Tm^3;

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(numer(t^2-t2Tm),t,Tm,n,true)): end:

Tmser(5);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

t2Zm:=subs(Tm=Zm^2,t2Tm);

LCQqKiMiIiIiJGMjRiUsJiokKUkjWm1HNiIiIiNGJUYlRiUhIiJGJSksJkYoRiUiIiRGJUYwRiUpRioiIidGLUYl

algTm:=numer(t^2-t2Tm); algZm:=subs(Tm=Zm^2,algTm);

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

S1stZm:=32*Zm^3*(Zm^3-3*Zm^2+3*Zm+3)/((Zm+1)*(Zm^2+3)^3);

LCQqLCIjSyIiIilJI1ptRzYiIiIkRiUsKiokRiZGJUYlKiZGKUYlKUYnIiIjRiUhIiIqJkYpRiVGJ0YlRiVGKUYlRiUsJkYnRiVGJUYlRi8pLCYqJEYtRiVGJUYpRiVGKUYvRiU=

eqS:=-Sx*(Sx-1)*(Sx-2)*(t*x^2+2*t*x+t-x)*(t*x^2-2*t*x+t-x)/x^2+t*(3*S1^2*t*x^2+2*t*x^4+3*S1*x^2-x^3+2*t-x)*Sx/x^2-4*x^2*t^2+2*x*t-3*S1*t*(S1*t+1);

LCwqLkkjU3hHNiIiIiIsJkYkRiZGJiEiIkYmLCZGJEYmIiIjRihGJiwqKiZJInRHRiVGJilJInhHRiVGKkYmRiYqKEYqRiZGLUYmRi9GJkYmRi1GJkYvRihGJiwqRixGJkYwRihGLUYmRi9GKEYmRi5GKEYoKipGLUYmLC4qKiIiJEYmKUkjUzFHRiVGKkYmRi1GJkYuRiZGJiooRipGJkYtRiYpRi8iIiVGJkYmKihGNUYmRjdGJkYuRiZGJiokKUYvRjVGJkYoKiZGKkYmRi1GJkYmRi9GKEYmRiRGJkYuRihGJiooRjpGJkYuRiYpRi1GKkYmRihGMEYmKipGNUYmRjdGJkYtRiYsJiomRjdGJkYtRiZGJkYmRiZGJkYo

<Text-field style="Heading 2" layout="Heading 2">Back to S(x); parametrization (series N)</Text-field>

Sser(4);

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An equation between S(x), t, x and Zm

algSZmt:=collect(numer(factor(subs(S1=t*S1stZm,eqS))),Sx,factor):nops(factor(%));

IiQlSA==

We can decrease the degree in t using the expression of t^2 in terms of Zm

t2Zm:=subs(Tm=Zm^2,t2Tm);

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algSZmsmall:=collect(op(4,factor(rem(algSZmt,numer(t^2-t2Zm),t))),[Sx,Zm],factor);

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

We note that this is in fact an equation over Q(t,x,Tm)

algSTmsmall:=subs(Zm=sqrt(Tm),algSZmsmall);

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n:=10: normal(series(subs(Tm=Tmser(n),Sx=Sser(n),algSTmsmall),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Let us check that S(x) has degree 12 over Q(x,t)

res:=numer(factor(resultant(algSTmsmall,algTm,Tm))): nops(%);

IiQzJg==

degree(res,Sx);

IiM3

S(xt) is even in t

n:=10: series(subs(x=x*t,Sser(n+1)),t,n+1);

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res:=numer(factor(subs(x=x*t,Sx=Sxt,t=sqrt(t2Tm),algSTmsmall))): nops(res);

IiIk

op(1,res); op(2,res);

LCZJI1RtRzYiIiIiRiUhIiI=

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nops(op(3,res));

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This is the equation satisfied by S(xt) over Q(x,Tm)

algSxtTm:=collect(op(3,res),Sxt,factor);

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Parametrisation of x

xTmUm:=factor(solve(16*Tm^2*(Um^2-Tm)-x*(-2*Tm+Um+Um*Tm)*(-9*Tm+Um^2+8*Tm*Um+Tm^2-Tm*Um^2),x));

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Expansion of Um in terms of (Tm-1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

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Let us check

n:=9:normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),numer(x-xTmUm)),t,n+1));

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res:=numer(factor(subs(x=xTmUm,algSxtTm))):nops(%);indets(%%);

IiIl

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op(1,res);op(2,res);

IiZPYic=

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degree(op(3,res),Sxt);degree(op(4,res),Sxt);

IiIi

IiIj

The quadratic term does not vanish

n:=10: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Sxt=subs(x=x*t,Sser(n)),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=20: normal(series(subs(Um=UmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Sxt=subs(x=x*t,Sser(n)),op(4,res)),t,n+1));

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SxtTmUm:=factor(solve(op(3,res),Sxt));

LCQqKCwmKiQpSSNVbUc2IiIiIyIiIiEiIkkjVG1HRihGKkYqLC4qKCIiJEYqKUYsRilGKkYmRipGKiokKUYsRi9GKkYqKigiIztGKkYwRipGJ0YqRisqJiIjQ0YqRjBGKkYqKiZGL0YqRiZGKkYrKiYiIipGKkYsRipGK0YqLDYqJkYwRiopRiciIiVGKkYqKihGKUYqRjJGKkYmRipGKyokKUYsRj1GKkYqKihGNEYqRjJGKkYnRipGKiooRjZGKkYwRipGJkYqRisqJEY8RipGKyomRjZGKkYyRipGKyooIiNbRipGMEYqRidGKkYqKigiIidGKkYsRipGJkYqRisqJkY5RipGMEYqRitGK0Yr

Let's check

n:=18: normal(series(subs(Um=convert(UmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),SxtTmUm),t,n+1));

KzFJInRHNiJJInhHRiQiIiMsJiokKUYlRiYiIiJGKiomRiZGKkYlRipGKiIiJSwoKiQpRiUiIiRGKkYqKiYiIidGKkYpRipGKiomIiM2RipGJUYqRipGMiwqKiQpRiVGLEYqRioqJiIjN0YqRi9GKkYqKiYiI1tGKkYpRipGKiomIiMmKUYqRiVGKkYqIiIpLCwqJClGJSIiJkYqRioqJiIjP0YqRjdGKkYqKiYiJEsiRipGL0YqRioqJiIkXSVGKkYpRipGKiomIiQjeUYqRiVGKkYqIiM1LC4qJClGJUYyRipGKiomIiNJRipGQUYqRioqJiIkIUhGKkY3RipGKiomIiUhXCJGKkYvRipGKiomIiVoWUYqRilGKkYqKiYiJS8hKUYqRiVGKkYqRjktSSJPRyUqcHJvdGVjdGVkRzYjRioiIzk=

normal(series(subs(x=x*t,Sser(10)),t,10));

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op(Sser);

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJXN1YnNHRik2JS9JInhHRiUiIiEvSSJ5R0YlRjAqKC1JJU5zZXJHRiVGIyIiIkYzRjdJInRHRiVGN0Y4RiVGJUYl

N0xtTmUm:=factor(SxtTmUm/xTmUm/t2Tm);

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

let us check this

n:=18: normal(series(subs(Um=convert(UmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),Tm=Tmser(n),N0xtTmUm),t,n+1));

KzNJInRHNiIiIiIiIiEsJkkieEdGJEYlIiIjRiVGKSwoKiQpRihGKUYlRiUqJiIiJ0YlRihGJUYlIiM2RiUiIiUsKiokKUYoIiIkRiVGJSomIiM3RiVGLEYlRiUqJiIjW0YlRihGJUYlIiMmKUYlRi4sLCokKUYoRjBGJUYlKiYiIz9GJUYzRiVGJSomIiRLIkYlRixGJUYlKiYiJF0lRiVGKEYlRiUiJCN5RiUiIiksLiokKUYoIiImRiVGJSomIiNJRiVGPEYlRiUqJiIkIUhGJUYzRiVGJSomIiUhXCJGJUYsRiVGJSomIiVoWUYlRihGJUYlIiUvISlGJSIjNSwwKiQpRihGLkYlRiUqJiIjVUYlRkdGJUYlKiYiJGImRiVGPEYlRiUqJiIlK1JGJUYzRiVGJSomIiZSdSJGJUYsRiVGJSomIiYlcF5GJUYoRiVGJSImVyEpKUYlRjYtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzk=

normal(series(subs(x=0,y=x*t,Nser(10)),t,10));

Ky9JInRHNiIiIiIiIiEsJkkieEdGJEYlIiIjRiVGKSwoKiQpRihGKUYlRiUqJiIiJ0YlRihGJUYlIiM2RiUiIiUsKiokKUYoIiIkRiVGJSomIiM3RiVGLEYlRiUqJiIjW0YlRihGJUYlIiMmKUYlRi4sLCokKUYoRjBGJUYlKiYiIz9GJUYzRiVGJSomIiRLIkYlRixGJUYlKiYiJF0lRiVGKEYlRiUiJCN5RiUiIiktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

xTmUm:=-16*Tm^2*(-Um^2+Tm)/((Tm*Um-2*Tm+Um)*(-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm));

LCQqLCIjOyIiIilJI1RtRzYiIiIjRiUsJiokKUkjVW1HRihGKUYlISIiRidGJUYlLCgqJkYnRiVGLUYlRiUqJkYpRiVGJ0YlRi5GLUYlRi4sLComRidGJUYsRiVGLiokRiZGJUYlKigiIilGJUYnRiVGLUYlRiVGK0YlKiYiIipGJUYnRiVGLkYuRi4=

Expansion of Um in terms of (Tm-1)

with(gfun): UmserTm:=proc(n) series(1+Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Um=1+Tm1*Um1,numer(x-xTmUm)),Tm1,Um1,n,true)),Tm1,n+2); end:

UmserTm(6);

KzNJJFRtMUc2IiIiIiIiISNGJSIiI0YlIyEiIiIiKUYoLCYqJiNGJSIjS0YlSSJ4R0YkRiVGJSNGJSIjO0YlIiIkLCgqJiMiIigiJEciRiVGMEYlRiojIiImRjhGKiomI0YlRjhGJSlGMEYoRiVGJSIiJSwqKiYjIiNQIiQ3JkYlRjBGJUYlI0Y3IiRjI0YlKiYjIiM2RkNGJUY9RiVGKiomI0YlRkNGJSlGMEYzRiVGJUY6LCwqJiNGJSIlWz9GJSlGMEY+RiVGJSMiI0AiJUM1RioqJiNGN0ZTRiVGS0YlRioqJiMiIyIpRk9GJUY9RiVGJSomI0ZIRjhGJUYwRiVGKiIiJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJUY3

SxtTmUm:=-(-Um^2+Tm)*(3*Tm^2*Um^2+Tm^3-16*Tm^2*Um+24*Tm^2-3*Um^2-9*Tm)/(Tm^2*Um^4-2*Tm^3*Um^2+Tm^4+16*Tm^3*Um-24*Tm^2*Um^2-Um^4-24*Tm^3+48*Tm^2*Um-6*Tm*Um^2-9*Tm^2);

N0xtTmUm := (16*(3*Tm^2*Um^2+Tm^3-16*Tm^2*Um+24*Tm^2-3*Um^2-9*Tm))*Tm*(Tm*Um-2*Tm+Um)*(-Tm*Um^2+Tm^2+8*Tm*Um+Um^2-9*Tm)/((Tm^2*Um^4-2*Tm^3*Um^2+Tm^4+16*Tm^3*Um-24*Tm^2*Um^2-Um^4-24*Tm^3+48*Tm^2*Um-6*Tm*Um^2-9*Tm^2)*(Tm-1)*(Tm+3)^3);

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

algSTmsmall := ((x-1)^2*(x+1)^2*Tm^4+(-512*t*x^3+8*x^4-512*t*x+240*x^2+8)*Tm^3+18*(x-1)^2*(x+1)^2*Tm^2-27*(x+1)^2*(x-1)^2)*Sx^3+(-3*(x-1)^2*(x+1)^2*Tm^4+(1536*t*x^3-24*x^4+1536*t*x-720*x^2-24)*Tm^3-54*(x-1)^2*(x+1)^2*Tm^2+81*(x+1)^2*(x-1)^2)*Sx^2+(-16*Tm^4*x^2-48*x*(16*t*x^2+16*t-11*x)*Tm^3-432*Tm^2*x^2+432*Tm*x^2)*Sx+4*x^2*(x^2+3)*Tm^4-16*x^2*(32*t*x-2*x^2+3)*Tm^3+72*x^2*(x^2+5)*Tm^2-432*Tm*x^2-108*x^2*(x-1)*(x+1);

<Text-field style="Heading 2" layout="Heading 2">Determination of N(x,0) (or R(x)) and parametrization</Text-field>

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYqRiopSSNEZUdGJyNGKkYsRipGKiooRixGKkkidEdGJ0YqLUkiUkdGJzYjIiIhRipGMCooIiIkRipGNUYqLUY3RihGKkYqRipGMCooRjtGKkY1RipGKUYwRio=

We first find an equation for DD:=S(x)-2S(1/x)+1

indets(algSTmsmall);

PCZJI1N4RzYiSSNUbUdGJEkidEdGJEkieEdGJA==

res:=numer(factor(resultant(subs(Sx=DD+2*S1x-1,algSTmsmall),subs(Sx=S1x,x=1/x,algSTmsmall),S1x))): nops(res);

IiIl

seq(nops(op(i,res)),i=1..nops(res));

NiYiIiIiIiMiI1IiJDUn

seq(degree(op(i,res),DD),i=1..nops(res));

NiYiIiFGIyIiJCIiJw==

n:=4: normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))+1,Tm=Tmser(n),op(3,res)),t,n+1));

Ky1JInRHNiIsJComIiVbPyIiIilJInhHRiQiIiNGKCEiIiIiISwkKihGJ0YoRipGKCwmKiRGKUYoRigiIilGLEYoRixGKCwoKiYiJnc8IkYoKUYqIiIlRihGKComIiZTZSRGKEYpRihGLCImKTNERixGKywkKigiJDcmRihGKkYsLCoqJkY3RigpRioiIidGKEYoKiYiI0ZGKEY2RihGKComIiR5JEYoRilGKEYsIiImRixGKEYsIiIkLUkiT0clKnByb3RlY3RlZEc2I0YoRjc=

n:=10: normal(series(subs(DD=Sser(n)-2*subs(x=1/x,Sser(n))+1,Tm=Tmser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

We decrease the degree in t

degree(op(4,res),t);

IiIk

res1:=factor(rem(op(4,res),numer(t^2-t2Tm),t)):nops(%);

IiQ6Jg==

algDDTm:=res1:

Hence DD has degree 6 over Q(x,T,t)

degree(algDDTm,DD);

IiIn

DD is in fact bicubic

seq(coeff(algDDTm,DD,2*i+1),i=0..2);

NiUiIiFGI0Yj

DD has degree 24 over Q(x,t)

res:=factor(resultant(algDDTm,numer(t^2-t2Tm),Tm)):nops(res);

IiVaUw==

degree(res,DD);

IiND

Back to R(x)

f13;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjBGKkYqRiopSSNEZUdGJyNGKkYsRipGKiooRixGKkkidEdGJ0YqLUkiUkdGJzYjIiIhRipGMCooIiIkRipGNUYqLUY3RihGKkYqRipGMCooRjtGKkY1RipGKUYwRio=

Recall that R0=S1 and that we have computed S1/t

S1stZm;

LCQqLCIjSyIiIilJI1ptRzYiIiIkRiUsKiokRiZGJUYlKiZGKUYlKUYnIiIjRiUhIiIqJkYpRiVGJ0YlRiVGKUYlRiUsJkYnRiVGJUYlRi8pLCYqJEYtRiVGJUYpRiVGKUYvRiU=

res:=numer(factor(subs(DD=(-3*t*Rx+2*t*S1+1-3*t/x)/sqrt(Delta),S1=t*S1stZm,Tm=Zm^2,algDDTm))): nops(%);

IiIj

op(1,res);nops(op(2,res));indets(res);

KiQpSSJ4RzYiIiIjIiIi

IiYzJkc=

PCZJI1J4RzYiSSNabUdGJEkidEdGJEkieEdGJA==

degree(op(2,res),Rx);degree(op(2,res),t);

IiIn

IiM4

Decreasing the degree in t

res:=numer(factor(rem(op(2,res),numer(t^2-t2Zm),t))): nops(res);

IiIm

seq(nops(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rx),i=1..nops(res));

NiciIiIiIiNGJEYkIiV1Ww==

NiciIiFGI0YjRiMiIic=

Equation of R(x) over Q(t,x,Z), of degree 6

algRZmt:=collect(op(5,res),Rx,factor):

degree(algRZmt,Rx);degree(algRZmt,t);

IiIn

IiIi

Odd powers of Z occur

nops(coeff(algRZmt,Zm,3));

IiNE

Equation for R(x) over Q(x,t): degree 24

res:=numer(factor(resultant(algRZmt,numer(t^2-t2Zm),Zm))): nops(%);

IiIm

seq(nops(op(i,res)),i=1..nops(res));seq(degree(op(i,res),Rx),i=1..nops(res));

NiciIiIiIiNGJCIlTkIiJUhF

NiciIiFGI0YjIiNDRiQ=

Choose the correct factor

n:=10:normal(series(subs(Rx=Rser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

n:=4:normal(series(subs(Rx=Rser(n),op(5,res)),t,n+1));

Ky9JInRHNiIsJComIiM7IiIiKUkieEdGJEYnRighIiIiIiEsJiomIiVHYEYoKUYqIiM8RihGKComRi9GKClGKiIjOkYoRihGKCwoKiYiJ1tAdkYoKUYqIiM9RihGKyomIig3LDsiRihGKUYoRisqJiInP2V2RigpRioiIzlGKEYrIiIjLCoqJiIpY2NHZUYoKUYqIiM+RihGKComIipvRShwNUYoRjBGKEYoKiYiKjsjZnY1RihGM0YoRigqJiIpWyZ6KWVGKClGKiIjOEYoRigiIiQsLComIispUUMmekVGKClGKiIjP0YoRisqJiIrS0JUM2VGKEY4RihGKyomIisncE12WShGKEYpRihGKyomIitPLWMqKmZGKEY+RihGKyomIitrKiozK0ZGKClGKiIjN0YoRisiIiUtSSJPRyUqcHJvdGVjdGVkRzYjRigiIiY=

algR:=collect(op(4,res),Rx,factor):

Parity of R(tx)/t

subs(x=x*t,Rser(3));

LCgqJiwmKiYpSSJ0RzYiIiIjIiIiKUkieEdGKEYpRipGKkYpRipGKilGJyIiJEYqRioqJkYsRipGLUYqRipGJ0Yq

res:=factor(subs(t=sqrt(t2Zm),factor(subs(x=x*t,Rx=Rxtst*t,algRZmt)))):indets(%);

PCVJJlJ4dHN0RzYiSSNabUdGJEkieEdGJA==

seq(indets(op(i,res)),i=1..nops(res));

Nio8IjwjSSNabUc2IkYkRiQ8JEYlSSJ4R0YmPCVJJlJ4dHN0R0YmRiVGKEYpRiQ=

Choose the correct factor

n:=10: normal(series(subs(Zm=Zmser(n),Rxtst=subs(x=x*t,Rser(n)/t),op(6,res)),t,n+1));

Ky1JInRHNiIsKComIilrKTNyJyIiIilJInhHRiQiIiNGKEYoKiYiKikpKSopPklGKEYqRighIiJGLUYoIiIhLCoqJkYtRigpRioiIiRGKEYuKiYiK2s/YkFNRihGKUYoRigqJiIsV2RodjkiRihGKkYoRi5GN0YoRissLComRi1GKClGKiIiJUYoRigqJkY3RihGMkYoRi4qJiIsb3ZWTmsqRihGKUYoRigqJiItKW9JQHUlSEYoRipGKEYuRkBGKEY7LCwqJkY3RihGOkYoRigqJkZARihGMkYoRi4qJiIuW2tMMm0+I0YoRilGKEYoKiYiLjd0KEdgQ2tGKEYqRihGLkZHRigiIictSSJPRyUqcHJvdGVjdGVkRzYjRigiIik=

n:=10: normal(series(subs(Zm=Zmser(n),Rxtst=subs(x=x*t,Rser(n)/t),op(7,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

algRxtstZm:=collect(op(7,res),Rxtst,factor);

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

degree

degree(algRxtstZm,Rxtst);

IiIk

And finally we parametrize x by Um

res:=numer(factor(subs(x=xZmUm,algRxtstZm))): nops(res);

IiIl

seq(indets(op(i,res)),i=1..nops(res)); seq(degree(op(i,res),Rxtst),i=1..nops(res));

NiY8IjwjSSNabUc2IjwlSSZSeHRzdEdGJkkjVW1HRiZGJUYn

NiYiIiFGIyIiIiIiIw==

We determine which term vanishes

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),Rxtst=1/t*subs(x=x*t,Rser(n)),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

n:=26: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),Rxtst=1/t*subs(x=x*t,Rser(n)),op(4,res)),t,n+1));

KytJInRHNiIsJiomIilrKTNyJyIiIkkieEdGJEYoRigiKkd4QE0iISIiIiM/LCoqJkYnRigpRikiIiRGKEYrKiYiKj9WYU4kRigpRikiIiNGKEYoKiYiK1c0WFVrRihGKUYoRigiLG8ienFBOUYrIiNBLCwqJiIqI2ZFOD9GKClGKSIiJUYoRisqJiIrZ3h6UmdGKEYvRihGKyomIiw7L11IWyRGKEYzRihGKComIi03UCdIXVwkRihGKUYoRigiLSFvalEicCUpRisiI0MtSSJPRyUqcHJvdGVjdGVkRzYjRigiI0U=

Rational expression of R(xt)/t in terms of Zm and Um

RxtstZmUm:=factor(solve(op(3,res),Rxtst));

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

op(Rser);

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSomSSJ0R0YlIiIiLUklc3Vic0dGKTYkL0kieUdGJSIiIS1JJU5zZXJHRiU2IywmRiRGLkYuISIiRi5GLUknbm9ybWFsR0YpRiVGJUYl

Nxt0ZmUm:=RxtstZmUm;

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),Nxt0ZmUm),t,n+1));

KzFJInRHNiIiIiIiIiEsJkkieEdGJEYlIiIjRiVGKSwoKiQpRihGKUYlRiUqJiIiJkYlRihGJUYlIiM2RiUiIiUsKiokKUYoIiIkRiVGJSomIiIqRiVGLEYlRiUqJiIjUEYlRihGJUYlIiMmKUYlIiInLCwqJClGKEYwRiVGJSomIiM5RiVGM0YlRiUqJiIjKClGJUYsRiVGJSomIiRLJEYlRihGJUYlIiQjeUYlIiIpLC4qJClGKEYuRiVGJSomIiM/RiVGPUYlRiUqJiIkcyJGJUYzRiVGJSomIiQ2KkYlRixGJUYlKiYiJVZMRiVGKEYlRiUiJS8hKUYlIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

n:=10: normal(series(subs(x=t*x,y=0,Nser(n)),t,n+1));

series(%%%-%,t,12);

IiIh

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

RxtstZmUm:=16*Zm^2*(Um*Zm^2-2*Zm^2+Um)*(Um^3*Zm^4+Um^2*Zm^5-Um*Zm^6-Zm^7-16*Um^2*Zm^4-16*Um*Zm^5-2*Um^3*Zm^2-2*Um^2*Zm^3+82*Um*Zm^4+18*Zm^5+16*Um^2*Zm^2+16*Um*Zm^3-128*Zm^4+Um^3+Um^2*Zm-17*Um*Zm^2-81*Zm^3)/((Zm-1)*(Zm+1)*(Zm^2+3)^3*(Um*Zm+Zm^2+Um-3*Zm)*(Um+Zm)*(Um*Zm-Zm^2-Um-3*Zm));

Nxt0ZmUm:=%;

<Text-field style="Heading 2" layout="Heading 2"><Font size="20">Back to the specializations of L and B at x=0 or y=0: rational expressions, and degree 24 for each</Font></Text-field>

We first compute the specialization at x=0 or y=0

Lxt0ZmUm:=factor(subs(Tm=Zm^2,(Mxt0TmUm+Nxt0ZmUm)/2));

LCQqMCIkYyMiIiIpSSNabUc2IiIiJUYlLCgqJkkjVW1HRihGJSlGJyIiI0YlRiUqJkYuRiVGLUYlISIiRixGJUYlLCZGLEYlRidGJUYwLCoqJkYsRiVGJ0YlRiUqJEYtRiVGJUYsRiUqJiIiJEYlRidGJUYwRjAsJkYnRiVGJUYwRjApLCZGNEYlRjZGJUY2RjBGJQ==

B0xtZmUm:=factor(subs(Tm=Zm^2,(Mxt0TmUm-Nxt0ZmUm)/2));

LCQqNCIjOyIiIiw4KiYpSSNVbUc2IiIiI0YlKUkjWm1HRioiIiRGJUYlKiQpRi0iIiZGJSEiIiomRihGJSlGLUYrRiVGMiooIiM3RiVGKUYlRixGJUYyKiZGLkYlKUYtIiIlRiVGMiomRihGJUYtRiVGMiooIiIpRiVGKUYlRjRGJUYlKiYiI0RGJUYsRiVGJSokRihGJUYlKihGOUYlRilGJUYtRiVGJSomIiM2RiVGNEYlRiVGJSwoKiZGKUYlRjRGJUYlKiZGK0YlRjRGJUYyRilGJUYlRjRGJSwqKiZGKUYlRi1GJUYlKiRGNEYlRjJGKUYyKiZGLkYlRi1GJUYyRjIsJkYpRiVGLUYlRjIsJkYtRiVGJUYyRjIsJkYtRiVGJUYlRjIpLCZGSEYlRi5GJUYuRjJGMg==

Let us check

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),Lxt0ZmUm),t,n+1));

KzFJInRHNiIiIiIiIiEsJiIiJEYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIidGJUYpRiVGJSIjQEYlIiIlLCoqJClGKUYoRiVGJSomIiM1RiVGLUYlRiUqJiIjXkYlRilGJUYlIiQkPkYlRi8sLCokKUYpRjFGJUYlKiYiIzpGJUY0RiVGJSomIiQxIkYlRi1GJUYlKiYiJDcmRiVGKUYlRiUiJU0/RiUiIiksLiokKUYpIiImRiVGJSomRjBGJUY8RiVGJSomIiQoPkYlRjRGJUYlKiYiJS43RiVGLUYlRiUqJiIlbmNGJUYpRiVGJSImJEdCRiVGNi1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

n:=10: normal(series(subs(x=x*t,y=0,Lser(n)),t,n+1));

series(%%%-%,t,12);

IiIh

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),B0xtZmUm),t,n+1));

KzFJInRHNiIiIiIiIiMsJiIjNUYlSSJ4R0YkRiUiIiUsKCokKUYpRiZGJUYlKiYiIzlGJUYpRiVGJSIkMyJGJSIiJywqKiQpRikiIiRGJUYlKiYiIz5GJUYtRiVGJSomIiQhPUYlRilGJUYlIiVfN0YlIiIpLCwqJClGKUYqRiVGJSomIiNERiVGNEYlRiUqJiIkI0hGJUYtRiVGJSomIiVDQkYlRilGJUYlIiZ6XyJGJUYoLC4qJClGKSIiJkYlRiUqJiIjS0YlRj5GJUYlKiYiJGQlRiVGNEYlRiUqJiIlVFVGJUYtRiVGJSomIiZFMCRGJUYpRiVGJSInV08+RiUiIzctSSJPRyUqcHJvdGVjdGVkRzYjRiVGLw==

n:=10: normal(series(subs(x=0,y=x*t,Bser(n)),t,n+1));

Ky9JInRHNiIiIiIiIiMsJiIjNUYlSSJ4R0YkRiUiIiUsKCokKUYpRiZGJUYlKiYiIzlGJUYpRiVGJSIkMyJGJSIiJywqKiQpRikiIiRGJUYlKiYiIz5GJUYtRiVGJSomIiQhPUYlRilGJUYlIiVfN0YlIiIpLCwqJClGKUYqRiVGJSomIiNERiVGNEYlRiUqJiIkI0hGJUYtRiVGJSomIiVDQkYlRilGJUYlIiZ6XyJGJUYoLUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

series(%%%-%,t,12);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

L0xtZmUm:=factor(subs(Tm=Zm^2,(M0xtZmUm+N0xtTmUm)/2));

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

Bxt0ZmUm:=factor(subs(Tm=Zm^2,(M0xtZmUm-N0xtTmUm)/2));

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

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),L0xtZmUm),t,n+1));

Ky9JInRHNiIiIiIiIiEsJiIiJEYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIilGJUYpRiVGJSIjQEYlIiIlLCoqJClGKUYoRiVGJSomIiM6RiVGLUYlRiUqJiIjeEYlRilGJUYlIiQkPkYlIiInLCwqJClGKUYxRiVGJSomIiNDRiVGNEYlRiUqJiIkIz5GJUYtRiVGJSomIiRJKUYlRilGJUYlIiVNP0YlRi8tSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

n:=10: normal(series(subs(x=0,y=x*t,Lser(n)),t,n+1));

KzFJInRHNiIiIiIiIiEsJiIiJEYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIilGJUYpRiVGJSIjQEYlIiIlLCoqJClGKUYoRiVGJSomIiM6RiVGLUYlRiUqJiIjeEYlRilGJUYlIiQkPkYlIiInLCwqJClGKUYxRiVGJSomIiNDRiVGNEYlRiUqJiIkIz5GJUYtRiVGJSomIiRJKUYlRilGJUYlIiVNP0YlRi8sLiokKUYpIiImRiVGJSomIiNORiVGPUYlRiUqJiIkJ1JGJUY0RiVGJSomIiVSQ0YlRi1GJUYlKiYiJUQnKkYlRilGJUYlIiYkR0JGJSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

series(%%%-%,t,12);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=16: normal(series(subs(Um=UmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),Bxt0ZmUm),t,n+1));

Ky1JInRHNiIiIiIiIiMsJiIjNUYlKiZGJkYlSSJ4R0YkRiVGJSIiJSwoKiYiIiRGJSlGKkYmRiVGJSomIiNIRiVGKkYlRiUiJDMiRiUiIicsKiomRitGJSlGKkYuRiVGJSomIiNnRiVGL0YlRiUqJiIkIVFGJUYqRiVGJSIlXzdGJSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJUYo

n:=10: normal(series(subs(x=x*t,y=0,Bser(n)),t,n+1));

Ky9JInRHNiIiIiIiIiMsJiIjNUYlKiZGJkYlSSJ4R0YkRiVGJSIiJSwoKiYiIiRGJSlGKkYmRiVGJSomIiNIRiVGKkYlRiUiJDMiRiUiIicsKiomRitGJSlGKkYuRiVGJSomIiNnRiVGL0YlRiUqJiIkIVFGJUYqRiVGJSIlXzdGJSIiKSwsKiYiIiZGJSlGKkYrRiVGJSomIiQxIkYlRjZGJUYlKiYiJFwqRiVGL0YlRiUqJiIla1xGJUYqRiVGJSImel8iRiVGKC1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

series(%%%-%,t,12);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Algebraic equations for the sections

We start with L(xt,0), and build its cubic equation over Q(x,Zm)

res:=factor(resultant(numer(Lxt0ZmUm-Lxt0),subs(Tm=Zm^2,numer(x-xTmUm)),Um)): nops(res);

IiIm

seq(nops(op(i,res)),i=1..nops(res));

NiciIiIiIiNGJEYkIiNw

seq(indets(op(i,res)),i=1..nops(res));

Nic8IjwjSSNabUc2IkYkRiQ8JUklTHh0MEdGJkYlSSJ4R0Ym

algLxt0Zm:=collect(op(5,res),Lxt0,factor);

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

Now with L(0,xt)

res:=factor(resultant(numer(L0xtZmUm-L0xt),subs(Tm=Zm^2,numer(x-xTmUm)),Um)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJE4i

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSVMMHh0R0YmRiVJInhHRiY=

algL0xtZm:=collect(op(6,res),L0xt,factor);

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

Now with B(xt,0)

res:=factor(resultant(numer(Bxt0ZmUm-Bxt0),subs(Tm=Zm^2,numer(x-xTmUm)),Um)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJGgk

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSVCeHQwR0YmRiVJInhHRiY=

algBxt0Zm:=collect(op(6,res),Bxt0,factor);

LCoqLilJInhHNiIiIiMiIiIpLCZJI1ptR0YmRihGKCEiIkYnRigpLCZGK0YoRihGKCIiJEYoKSwmKiQpRitGJ0YoRihGL0YoIiIqRigsRiomKUYrIiM7RigpRiUiIiVGKEYoKihGOEYoKUYrIiM5RihGOUYoRigqKCIkNyZGKEY8RigpRiVGL0YoRiwqKEY/RihGPEYoRiRGKEYsKigiJCsiRigpRisiIzdGKEY5RihGKCooIiUnNCVGKEZERihGQEYoRiwqKCImUzknRihGREYoRiRGKEYoKigiJClHRigpRisiIzVGKEY5RihGKCooIidzNThGKEZERihGJUYoRiwqKCIlOyMqRihGTEYoRkBGKEYsKiYiJk9iJ0YoRkRGKEYoKihGUUYoRkxGKEYkRihGLCooIiRxI0YoKUYrIiIpRihGOUYoRigqKCIkSyVGKClGKyIiJ0YoRjlGKEYsKigiJkNRIkYoRmVuRihGQEYoRigqKEZobkYoRmVuRihGJEYoRigqKCIkcypGKClGK0Y6RihGOUYoRiwqJiIkSChGKEY5RihGKEYoKUklQnh0MEdGJkYvRihGKCoyIiRvKEYoRmVuRihGJUYoRipGKClGLkYnRigpRjFGZm5GKEY1RigpRmBvRidGKEYoKi5GT0YoRkRGKEYuRigpRjFGL0YoLGZuRjZGKCooIiNLRigpRisiIzpGKEZARihGKEY7RigqKCIkUydGKEY8RihGQEYoRiwqKCIlTzpGKEY8RihGJEYoRiwqKCIkXyRGKClGKyIjOEYoRkBGKEYoRkJGKCooRkdGKEZjcEYoRiRGKEYsKigiJT9eRihGREYoRkBGKEYsKigiJSM+KUYoRmNwRihGJUYoRigqKCImIypcKUYoRkRGKEYkRihGKCooIiVXOEYoKUYrIiM2RihGQEYoRihGSkYoKigiJ1NRO0YoRkRGKEYlRihGLCooIiYpRzdGKEZecUYoRiRGKEYsKigiJj86IkYoRkxGKEZARihGLEZSRigqKCImd1gjRihGXnFGKEYlRihGKCooRmdxRihGTEYoRiRGKEYsKigiJUc8RigpRitGNEYoRkBGKEYoRlVGKCooRlFGKEZXRihGJEYoRigqKCIkaylGKClGKyIiKEYoRkBGKEYsRllGLCooIiYhRzxGKEZlbkYoRkBGKEYoRmluRigqKCIlI2YjRigpRisiIiZGKEZARihGLEZqbkYsRl1vRihGKEZgb0YoRigqKiIrQz11dDVGKClGKyIjQEYoRipGKCxIKiZGTEYoRiRGKEYoKihGJ0YoRltyRihGJEYoRiwqKEZqb0YoRltyRihGJUYoRiwqKEZmckYoRldGKEYkRihGKCooIiQjPkYoRldGKEYlRihGLCooIiM/RihGX3JGKEYkRihGLComIiRjI0YoRldGKEYoKigiJCVRRihGX3JGKEYlRihGKCooRk1GKEZlbkYoRiRGKEYsKiZGP0YoRl9yRihGLCooIiR3JkYoRmVuRihGJUYoRigqKCIjc0YoRmVyRihGJEYoRiwqJkZlc0YoRmVuRihGLCooRktGKEZlckYoRiVGKEYoKigiIyEqRihGXG9GKEYkRihGLCooIiQzIkYoKUYrRi9GKEYkRihGLCooIiROIkYoRjNGKEYkRihGLCooIiNhRihGK0YoRiRGKEYsKiYiI0ZGKEYkRihGLEYoRig=

Now with B(0,xt)

res:=factor(resultant(numer(B0xtZmUm-B0xt),subs(Tm=Zm^2,numer(x-xTmUm)),Um)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJGUi

seq(indets(op(i,res)),i=1..nops(res));

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSVCMHh0R0YmRiVJInhHRiY=

algB0xtZm:=collect(op(6,res),B0xt,factor);

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

Degrees over Q(x,t): 24 for each series

algZm;

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

res:=factor(resultant(algLxt0Zm,algZm,Zm)): nops(res);

IiVTNw==

degree(res,Lxt0);

IiND

res:=factor(resultant(algL0xtZm,algZm,Zm)): nops(res);

IiVUQg==

degree(res,L0xt);

IiND

res:=factor(resultant(algBxt0Zm,algZm,Zm)): nops(res);

IiVmTw==

degree(res,Bxt0);

IiND

res:=factor(resultant(algB0xtZm,algZm,Zm)): nops(res);

IiVoPg==

degree(res,B0xt);

IiND

results of this subsection

Lxt0ZmUm:=256*Zm^4*(Um*Zm^2-2*Zm^2+Um)/((Um+Zm)*(Um*Zm+Zm^2+Um-3*Zm)*(Zm-1)*(Zm^2+3)^3);

L0xtZmUm:=-(16*(Um*Zm+Zm^2+Um-3*Zm))*(Um^2*Zm^2-Zm^4-4*Um*Zm^2+Um^2+3*Zm^2)*(Um*Zm-Zm^2-Um-3*Zm)^2*Zm^2*(Um*Zm^2-2*Zm^2+Um)/((Zm-1)*(Zm^2+3)^3*(Um+Zm)*(Um^4*Zm^4-2*Um^2*Zm^6+Zm^8+16*Um*Zm^6-24*Um^2*Zm^4-24*Zm^6+48*Um*Zm^4-Um^4-6*Um^2*Zm^2-9*Zm^4));

A slightly smaller expression if we multiply by x

factor(xZmUm*L0xtZmUm);

LCQqMiIkYyMiIiIpSSNabUc2IiIiJ0YlLCZJI1VtR0YoRiVGJyEiIkYlLCoqJkYrRiVGJ0YlRiUqJClGJyIiI0YlRixGK0YsKiYiIiRGJUYnRiVGLEYlLCwqJilGK0YxRiVGMEYlRiUqJClGJyIiJUYlRiwqKEY5RiVGK0YlRjBGJUYsKiRGNkYlRiUqJkYzRiVGMEYlRiVGJSwmRidGJUYlRixGLCksJkYvRiVGM0YlRjNGLCw2KiYpRitGOUYlRjhGJUYlKihGMUYlRjZGJUYmRiVGLCokKUYnIiIpRiVGJSooIiM7RiVGK0YlRiZGJUYlKigiI0NGJUY2RiVGOEYlRiwqJkZKRiVGJkYlRiwqKCIjW0YlRitGJUY4RiVGJSokRkJGJUYsKihGKUYlRjZGJUYwRiVGLComIiIqRiVGOEYlRixGLEYl

collect(Um^4*Zm^4-2*Um^2*Zm^6+Zm^8+16*Um*Zm^6-24*Um^2*Zm^4-24*Zm^6+48*Um*Zm^4-Um^4-6*Um^2*Zm^2-9*Zm^4,Um,factor);

LCoqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsJiokKUYlIiIjRidGJ0YnRidGJylJI1VtR0YmIiIlRidGJyoqRi1GJ0YsRicsKCokKUYlRjBGJ0YnKiYiIzdGJ0YsRidGJyIiJEYnRicpRi9GLUYnRigqKiIjO0YnRjRGJywmRitGJ0Y3RidGJ0YvRidGJyomRjRGJywoRjNGJyomIiNDRidGLEYnRigiIipGKEYnRic=

collect(%,Zm,factor);

LCwqJClJI1ptRzYiIiIpIiIiRigqKiIiI0YoLCZJI1VtR0YmRihGKiEiIkYoLCZGLEYoIiInRi1GKClGJUYvRihGLSooLCZGLEYoIiIkRi1GKCwqKiQpRixGM0YoRigqJkYzRigpRixGKkYoRigqJiIjOkYoRixGKEYtRjNGKEYoKUYlIiIlRihGKCooRi9GKEY4RigpRiVGKkYoRi0qJClGLEY8RihGLQ==

Bxt0ZmUm:=32*(Um*Zm^2-2*Zm^2+Um)^2*Zm^2*(Um*Zm-Zm^2-Um-3*Zm)*(Um*Zm+Zm^2+Um-3*Zm)*(Um^2*Zm^2+2*Um*Zm^3+Zm^4-2*Um*Zm^2-6*Zm^3-Um^2-3*Zm^2)/((Um^4*Zm^4-2*Um^2*Zm^6+Zm^8+16*Um*Zm^6-24*Um^2*Zm^4-24*Zm^6+48*Um*Zm^4-Um^4-6*Um^2*Zm^2-9*Zm^4)*(Um+Zm)*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

Again, it's smaller if we multiply by x

factor(xZmUm*Bxt0ZmUm);

LCQqNCIkNyYiIiIpSSNabUc2IiIiJ0YlLCZJI1VtR0YoRiVGJyEiIkYlLCgqJkYrRiUpRiciIiNGJUYlKiZGMEYlRi9GJUYsRitGJUYlLDAqJilGK0YwRiVGL0YlRiUqKEYwRiVGK0YlKUYnIiIkRiVGJSokKUYnIiIlRiVGJSooRjBGJUYrRiVGL0YlRiwqJkYpRiVGNkYlRiwqJEY0RiVGLComRjdGJUYvRiVGLEYlLDYqJilGK0Y6RiVGOUYlRiUqKEYwRiVGNEYlRiZGJUYsKiQpRiciIilGJUYlKigiIztGJUYrRiVGJkYlRiUqKCIjQ0YlRjRGJUY5RiVGLComRklGJUYmRiVGLCooIiNbRiVGK0YlRjlGJUYlKiRGQUYlRiwqKEYpRiVGNEYlRi9GJUYsKiYiIipGJUY5RiVGLEYsLCZGJ0YlRiVGLEYsLCZGJ0YlRiVGJUYsKSwmKiRGL0YlRiVGN0YlRjdGLEYs

collect(Um^2*Zm^2+2*Um*Zm^3+Zm^4-2*Um*Zm^2-6*Zm^3-Um^2-3*Zm^2,Zm,factor);

LCoqJClJI1ptRzYiIiIlIiIiRigqJiwmKiYiIiNGKEkjVW1HRiZGKEYoIiInISIiRigpRiUiIiRGKEYoKigsJkYtRihGKEYoRigsJkYtRihGMUYvRigpRiVGLEYoRigqJClGLUYsRihGLw==

B0xtZmUm:=-(16*(Um^2*Zm^3-Zm^5-Um^2*Zm^2-12*Um*Zm^3-3*Zm^4-Um^2*Zm+8*Um*Zm^2+25*Zm^3+Um^2+4*Um*Zm+11*Zm^2))*(Um*Zm^2-2*Zm^2+Um)*Zm^2/((Um*Zm-Zm^2-Um-3*Zm)*(Um+Zm)*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

collect(Um^2*Zm^3-Zm^5-Um^2*Zm^2-12*Um*Zm^3-3*Zm^4-Um^2*Zm+8*Um*Zm^2+25*Zm^3+Um^2+4*Um*Zm+11*Zm^2,Um,factor);

LCgqKCwmSSNabUc2IiIiIkYnRidGJyksJkYlRidGJyEiIiIiI0YnKUkjVW1HRiZGK0YnRicqLCIiJUYnRiVGJywmKiYiIiRGJ0YlRidGJ0YnRidGJ0YpRidGLUYnRioqJilGJUYrRicsKiokKUYlRjJGJ0YnKiZGMkYnRjRGJ0YnKiYiI0RGJ0YlRidGKiIjNkYqRidGKg==

Here we can recover the series G(xt,0) in gessel's model (Section 6)

Gxt0ZmUm:=factor(Lxt0ZmUm-B0xtZmUm);

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

subs(Zm=sqrt(Tm),expand((Um*Zm-Zm^2-Um-3*Zm)*(Um*Zm+Zm^2+Um-3*Zm)));

LCwqJkkjVG1HNiIiIiIpSSNVbUdGJSIiI0YmRiYqJClGJEYpRiYhIiIqKCIiKUYmRiRGJkYoRiZGLCokRidGJkYsKiYiIipGJkYkRiZGJg==

collect(Um^3*Zm^4+Um^2*Zm^5-Um*Zm^6-Zm^7-16*Um^2*Zm^4-16*Um*Zm^5-2*Um^3*Zm^2-2*Um^2*Zm^3+82*Um*Zm^4+18*Zm^5+16*Um^2*Zm^2+16*Um*Zm^3-128*Zm^4+Um^3+Um^2*Zm-17*Um*Zm^2-81*Zm^3,Um,factor);

LCoqKCksJkkjWm1HNiIiIiJGKCEiIiIiI0YoKSwmRiZGKEYoRihGKkYoKUkjVW1HRiciIiRGKEYoKixGJkYoRiVGKEYsRigsKCokKUYmRipGKEYoKiYiIztGKEYmRihGKUYoRilGKClGLkYqRihGKCooRjNGKCwsKiQpRiYiIiVGKEYoKiZGNUYoKUYmRi9GKEYoKiYiIyMpRihGM0YoRilGNEYpIiM8RihGKEYuRihGKSomRj1GKCwqRjlGKComIiM9RihGM0YoRikqJiIkRyJGKEYmRihGKCIjIilGKEYoRik=

Later we'll need the values at x=y=0. When x=0, we have Um=Zm

numer(xZmUm);numer(t2Zm);

LCQqKiIjOyIiIilJI1ptRzYiIiIlRiUsJkkjVW1HRihGJUYnISIiRiUsJkYrRiVGJ0YlRiVGLA==

KiYsJiokKUkjWm1HNiIiIiMiIiJGKUYpISIiRikpLCZGJEYpIiIkRilGLUYp

UmserTm(2);Zmser(2);

KytJJFRtMUc2IiIiIiIiISNGJSIiI0YlIyEiIiIiKUYoLUkiT0clKnByb3RlY3RlZEc2I0YlIiIk

KydJInRHNiIiIiIiIiEtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIiM=

L00Zm:=factor(subs(Um=Zm,L0xtZmUm)); B00Zm:=factor(subs(Um=Zm,B0xtZmUm));

LCQqKCIjayIiIilJI1ptRzYiIiIkRiUpLCYqJClGJyIiI0YlRiVGKUYlRikhIiJGJQ==

LCQqLiIjSyIiIiwmSSNabUc2IkYlRiUhIiJGJSwoKiQpRiciIiNGJUYlKiZGLUYlRidGJUYpRiVGKUYlKUYnIiIkRiUsJkYnRiVGJUYlRikpLCZGK0YlRjBGJUYwRilGKQ==

Let us check

n:=10: normal(series(subs(Zm=Zmser(n),L00Zm),t,n+1));

Ky9JInRHNiIiIiIiIiEiIiQiIiMiI0AiIiUiJCQ+IiInIiVNPyIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

subs(x=0,y=0,Lser(n));

LC4qJiImJEdCIiIiKUkidEc2IiIjNUYlRiUqJiIlTT9GJSlGJyIiKUYlRiUqJiIkJD5GJSlGJyIiJ0YlRiUqJiIjQEYlKUYnIiIlRiVGJSomIiIkRiUpRiciIiNGJUYlRiVGJQ==

n:=10: normal(series(subs(Zm=Zmser(n),B00Zm),t,n+1));

Ky1JInRHNiIiIiIiIiMiIzUiIiUiJDMiIiInIiVfNyIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJUYn

subs(x=0,y=0,Bser(n));

LCwqJiImel8iIiIiKUkidEc2IiIjNUYlRiUqJiIlXzdGJSlGJyIiKUYlRiUqJiIkMyJGJSlGJyIiJ0YlRiUqJkYpRiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

Algebraic equations (over Q(x,Z)) for these specializations

algLxt0Zm := x*(Zm-1)*(Zm+1)*(Zm^2+3)^9*Lxt0^3+(Zm^2+3)^6*(Zm^8*x^2-32*Zm^7*x+8*Zm^6*x^2-128*Zm^6*x+256*Zm^6-192*Zm^5*x+18*Zm^4*x^2+96*Zm^3*x-27*x^2)*Lxt0^2-64*Zm^3*(Zm^2+3)^3*(Zm^8*x^2-96*Zm^7*x+8*Zm^6*x^2-128*Zm^6*x+256*Zm^6-192*Zm^5*x+18*Zm^4*x^2-96*Zm^3*x-27*x^2)*Lxt0-262144*Zm^9*x*(Zm^2+1)^2;

algL0xtZm := x^2*(Zm-1)^2*(Zm+1)^2*(Zm^2+3)^9*(Zm^16*x^4+16*Zm^14*x^4-512*Zm^14*x^3-512*Zm^14*x^2+100*Zm^12*x^4-4096*Zm^12*x^3+61440*Zm^12*x^2+288*Zm^10*x^4-131072*Zm^12*x-9216*Zm^10*x^3+65536*Zm^12-9216*Zm^10*x^2+270*Zm^8*x^4-432*Zm^6*x^4+13824*Zm^6*x^3+13824*Zm^6*x^2-972*Zm^4*x^4+729*x^4)*L0xt^3-65536*Zm^12*(Zm^2+3)^3*(Zm^16*x^4+64*Zm^15*x^3+16*Zm^14*x^4-256*Zm^14*x^3+1536*Zm^14*x^2+704*Zm^13*x^3+100*Zm^12*x^4-8192*Zm^13*x^2-2048*Zm^12*x^3+16384*Zm^13*x+14336*Zm^12*x^2+2688*Zm^11*x^3+288*Zm^10*x^4-65536*Zm^12*x-24576*Zm^11*x^2-4608*Zm^10*x^3+65536*Zm^12+49152*Zm^11*x+21504*Zm^10*x^2+3456*Zm^9*x^3+270*Zm^8*x^4-18432*Zm^8*x^2-1728*Zm^7*x^3-432*Zm^6*x^4+6912*Zm^6*x^3+13824*Zm^6*x^2-5184*Zm^5*x^3-972*Zm^4*x^4+729*x^4)*L0xt-1073741824*Zm^21*(Zm^8*x^2+8*Zm^6*x^2-256*Zm^6+18*Zm^4*x^2-27*x^2);

algBxt0Zm := x^2*(Zm-1)^2*(Zm+1)^3*(Zm^2+3)^9*(Zm^16*x^4+16*Zm^14*x^4-512*Zm^14*x^3-512*Zm^14*x^2+100*Zm^12*x^4-4096*Zm^12*x^3+61440*Zm^12*x^2+288*Zm^10*x^4-131072*Zm^12*x-9216*Zm^10*x^3+65536*Zm^12-9216*Zm^10*x^2+270*Zm^8*x^4-432*Zm^6*x^4+13824*Zm^6*x^3+13824*Zm^6*x^2-972*Zm^4*x^4+729*x^4)*Bxt0^3+768*Zm^6*x*(Zm-1)*(Zm+1)^2*(Zm^2+3)^6*(Zm^16*x^4+16*Zm^14*x^4-512*Zm^14*x^3-512*Zm^14*x^2+100*Zm^12*x^4-4096*Zm^12*x^3+61440*Zm^12*x^2+288*Zm^10*x^4-131072*Zm^12*x-9216*Zm^10*x^3+65536*Zm^12-9216*Zm^10*x^2+270*Zm^8*x^4-432*Zm^6*x^4+13824*Zm^6*x^3+13824*Zm^6*x^2-972*Zm^4*x^4+729*x^4)*Bxt0^2+131072*Zm^12*(Zm+1)*(Zm^2+3)^3*(Zm^16*x^4+32*Zm^15*x^3+16*Zm^14*x^4-640*Zm^14*x^3-1536*Zm^14*x^2+352*Zm^13*x^3+100*Zm^12*x^4-4096*Zm^13*x^2-5120*Zm^12*x^3+8192*Zm^13*x+84992*Zm^12*x^2+1344*Zm^11*x^3+288*Zm^10*x^4-163840*Zm^12*x-12288*Zm^11*x^2-11520*Zm^10*x^3+65536*Zm^12+24576*Zm^11*x-24576*Zm^10*x^2+1728*Zm^9*x^3+270*Zm^8*x^4+9216*Zm^8*x^2-864*Zm^7*x^3-432*Zm^6*x^4+17280*Zm^6*x^3+13824*Zm^6*x^2-2592*Zm^5*x^3-972*Zm^4*x^4+729*x^4)*Bxt0+1073741824*Zm^21*(Zm-1)*(Zm^10*x^2-2*Zm^9*x^2-32*Zm^9*x+5*Zm^8*x^2-192*Zm^8*x-20*Zm^7*x^2+256*Zm^8+384*Zm^7*x-10*Zm^6*x^2-512*Zm^7+576*Zm^6*x-72*Zm^5*x^2-256*Zm^6+288*Zm^5*x-90*Zm^4*x^2-108*Zm^3*x^2-135*Zm^2*x^2-54*Zm*x^2-27*x^2);

algB0xtZm := x^2*(Zm-1)^2*(Zm+1)^3*(Zm^2+3)^9*B0xt^3+x*(Zm-1)*(Zm+1)^2*(Zm^2+3)^6*(Zm^8*x^2+32*Zm^7*x+8*Zm^6*x^2-128*Zm^6*x-512*Zm^6+192*Zm^5*x+18*Zm^4*x^2-96*Zm^3*x-27*x^2)*B0xt^2-64*Zm^3*(Zm+1)*(Zm^2+3)^3*(8*Zm^11*x^2-Zm^10*x^3+256*Zm^10*x-32*Zm^9*x^2-7*Zm^8*x^3-1024*Zm^9*x+128*Zm^8*x^2-1024*Zm^9+1280*Zm^8*x+48*Zm^7*x^2-10*Zm^6*x^3-128*Zm^6*x^2-512*Zm^6*x+96*Zm^5*x^2+18*Zm^4*x^3-120*Zm^3*x^2+27*Zm^2*x^3-27*x^3)*B0xt+16384*Zm^9*(Zm-1)*(4*Zm^9*x-Zm^8*x^2+8*Zm^8*x-2*Zm^7*x^2+128*Zm^8-52*Zm^7*x+6*Zm^6*x^2-256*Zm^7+16*Zm^6*x+14*Zm^5*x^2-128*Zm^6+60*Zm^5*x+12*Zm^4*x^2-24*Zm^4*x+10*Zm^3*x^2-12*Zm^3*x-6*Zm^2*x^2-22*Zm*x^2-11*x^2);

<Text-field style="Heading 2" layout="Heading 2">Degrees: 72 for L(x,y), B(x,y) and P(x,y)</Text-field>

eqL;

L(xt,yt) is even in t

n:=8:normal(series(subs(x=t*x,y=t*y,Lser(n)),t,n+1));

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

LCYiIiJGIyomSSJ0RzYiRiMsKkkieEdGJkYjKiRGKCEiIkYjSSJ5R0YmRiMqJEYrRipGI0YjRio=

An equation for L(xt,yt) in terms of Zm, Um and Umy (the counterpart of Um with y instead of x)

eqLxtyt:=numer(factor(collect(subs(Ker=K,x=x*t,y=y*t,L(0,y*t)=L0yt,L(x*t,0)=Lxt0,L(x*t,y*t)=Lxtyt,B(0,y*t)=B0yt,t=sqrt(t2Zm),L(0,0)=L00Zm,B(0,0)=B00Zm,Lxt0=Lxt0ZmUm, L0yt=subs(Um=Umy,L0xtZmUm), B0yt=subs(Um=Umy,B0xtZmUm), eqL),[L,B],factor))):

nops(%);indets(%%);

IiVwRw==

PChJJkx4dHl0RzYiSSNVbUdGJEkkVW15R0YkSSNabUdGJEkieEdGJEkieUdGJA==

We take x=2 and y=-2 and compute the degree of L(xt,yt). First, equations for Um and Umy

algUmx2:=factor(subs(Tm=Zm^2,numer(2-xTmUm)));

LDQqKCIiIyIiIilJI1VtRzYiIiIkRiUpSSNabUdGKCIiJUYlISIiKihGJEYlRidGJSlGKyIiJ0YlRiUqKEYsRiUpRidGJEYlRipGJUYlKiYiIzdGJUYvRiVGJSooIiNbRiVGJ0YlRipGJUYtKihGNEYlRjJGJSlGK0YkRiVGJSomIiNPRiVGKkYlRiUqJkYkRiVGJkYlRiUqKCIjPUYlRidGJUY4RiVGLQ==

algUmym2:=factor(subs(Tm=Zm^2,Um=Umy,numer(-2-xTmUm)));

LDQqKCIiIyIiIilJJFVteUc2IiIiJEYlKUkjWm1HRigiIiVGJUYlKihGJEYlRidGJSlGKyIiJ0YlISIiKigiI09GJSlGJ0YkRiVGKkYlRjAqJiIjP0YlRi5GJUYlKigiI1tGJUYnRiVGKkYlRiUqKCIjN0YlRjNGJSlGK0YkRiVGMComRjJGJUYqRiVGMComRiRGJUYmRiVGMCooIiM9RiVGJ0YlRjpGJUYl

eq2m2:=(factor(subs(x=2,y=-2,eqLxtyt))): nops(%);

IiIk

op(1,eq2m2); op(2,eq2m2);nops(op(3,eq2m2));

IiNr

KiQpSSNabUc2IiIiIyIiIg==

IiQpcA==

eq2m2bis:=op(3,eq2m2):

res:=factor(resultant(eq2m2bis,algUmym2,Umy)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJVQ1

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSZMeHR5dEdGJkkjVW1HRiZGJQ==

res1:=op(6,res):

res2:=factor(resultant(res1,algUmx2,Um)): nops(res2);

IiIm

seq(nops(op(i,res2)),i=1..nops(res2)); seq(indets(op(i,res2)),i=1..nops(res2));

NiciIiIiIiNGJEYkIiU4NQ==

Nic8IjwjSSNabUc2IkYkRiQ8JEkmTHh0eXRHRiZGJQ==

res3:=op(5,res2):

res4:=factor(resultant(res3,algZm,Zm)): nops(res4);

IiV6YA==

degree(res4,Lxtyt);

IiNz

n:=20: series(subs(Lxtyt=subs(x=2*t,y=-2*t,Lser(n)),res4),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNB

Now for B

eqB;

B(xt,yt) is even in t

n:=10:normal(series(subs(x=t*x,y=t*y,Bser(n)),t,n+1));

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

eqBxtyt:=numer(factor(collect(subs(Ker=K,x=x*t,y=y*t,B(0,y*t)=B0yt,L(x*t,0)=Lxt0,B(x*t,y*t)=Bxtyt,B(x*t,0)=Bxt0,t=sqrt(t2Zm),L(0,0)=L00Zm,B(0,0)=B00Zm,Lxt0=Lxt0ZmUm, B0yt=subs(Um=Umy,B0xtZmUm), Bxt0=Bxt0ZmUm, eqB),[L,B],factor))):

nops(%);indets(eqBxtyt);

IiVJSA==

PChJJkJ4dHl0RzYiSSNVbUdGJEkkVW15R0YkSSNabUdGJEkieEdGJEkieUdGJA==

We take x=2 and y=-2 and compute the degree of B(xt,yt)

eq2m2:=(factor(subs(x=2,y=-2,eqBxtyt))): nops(%);

IiIk

op(1,eq2m2); op(2,eq2m2);nops(op(3,eq2m2));

IiRHIg==

KiQpSSNabUc2IiIiIyIiIg==

IiRyKA==

eq2m2bis:=op(3,eq2m2):

res:=factor(resultant(eq2m2bis,algUmym2,Umy)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJUVQ

Nig8IjwjSSNabUc2IkYkRiRGJDwlSSZCeHR5dEdGJkkjVW1HRiZGJQ==

res1:=op(6,res):

res2:=factor(resultant(res1,algUmx2,Um)): nops(res2);

IiIn

seq(nops(op(i,res2)),i=1..nops(res2)); seq(indets(op(i,res2)),i=1..nops(res2));

NigiIiIiIiNGJEYkRiQiJTk1

Nig8IjwjSSNabUc2IkYkRiRGJDwkSSZCeHR5dEdGJkYl

res3:=op(6,res2):

res4:=factor(resultant(res3,algZm,Zm)): nops(res4);

IiUhUSY=

degree(res4,Bxtyt);

IiNz

Let us check this equation

n:=20: normal(series(subs(Bxtyt=subs(x=2*t,y=-2*t,Bser(n)),res4),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNB

Finally for P

subs(x=x*t,y=y*t,Pexpr);

LCoqKC1JIkJHNiI2JComSSJ0R0YmIiIiSSJ4R0YmRioiIiFGKkkieUdGJiEiIkYpRi5GLiooLUkiTEdGJjYkRigqJkYtRipGKUYqRipGKUYuRitGLkYqKigtRiVGMkYqRi1GLkYpRi5GKiooLUYxNiRGLEYzRipGKUYuRitGLkYu

P(xt,yt) is odd in t

n:=10:normal(series(subs(x=x*t,y=y*t,Pser(n)),t,n+1));

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

indets(eqBxtyt);

PChJJkJ4dHl0RzYiSSNVbUdGJEkkVW15R0YkSSNabUdGJEkieEdGJEkieUdGJA==

Equation for tP(xt,yt) when x=2 and y=-2

res:=factor(numer(subs(B(x*t,0)=Bxt0ZmUm, L(0,y*t)=subs(Um=Umy,L0xtZmUm), L(x*t,y*t)=solve(subs(x=2,y=-2,eqLxtyt),Lxtyt), B(x*t,y*t)=solve(subs(x=2,y=-2,eqBxtyt),Bxtyt),x=2,y=-2,Pxtytft-t*subs(x=x*t,y=y*t,Pexpr)))): nops(res);

IiM2

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

Ni0iIiIiIiNGJEYkIiIlIiM1RiRGJUYmRiQiJSpHIw==

Ni08I0kidEc2IjwjSSNabUdGJUYmRiY8JEkkVW15R0YlRidGKEYoPCRJI1VtR0YlRidGKkYqPCZJKFB4dHl0ZnRHRiVGK0YpRic=

eqP2m2:=op(11,res):degree(%,Pxtytft);

IiIi

res:=factor(resultant(eqP2m2,algUmym2,Umy)): nops(res);

IiIn

seq(nops(op(i,res)),i=1..nops(res)); seq(indets(op(i,res)),i=1..nops(res));

NigiIiIiIiNGJEYkRiQiJVVd

Nig8IjwjSSNabUc2IkYkRiRGJDwlSShQeHR5dGZ0R0YmSSNVbUdGJkYl

res1:=op(6,res):

res2:=factor(resultant(res1,algUmx2,Um)): nops(res2);

IiIn

seq(nops(op(i,res2)),i=1..nops(res2)); seq(indets(op(i,res2)),i=1..nops(res2));

NigiIiIiIiNGJEYkRiQiJUg4

Nig8IjwjSSNabUc2IkYkRiRGJDwkSShQeHR5dGZ0R0YmRiU=

res3:=op(6,res2):

res4:=factor(resultant(res3,algZm,Zm)): nops(res4);

IiVRISo=

degree(res4,Pxtytft);

IiNz

Let us check

n:=20: series(subs(Pxtytft=t*subs(x=2*t,y=-2*t,Pser(n)),res4),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNB

<Text-field style="Heading 2" layout="Heading 2">Walks ending at a prescribed position</Text-field>

Let us prove that the coefficient of x^i in L(xt,0) is a rational function in Z

algLxt0Zm;

series(subs(x=x*t,y=0,Lser(4)),t,5);

KytJInRHNiIiIiIiIiEsJiIiJEYlSSJ4R0YkRiUiIiMsKCokKUYpRipGJUYlKiYiIidGJUYpRiVGJSIjQEYlIiIlLUkiT0clKnByb3RlY3RlZEc2I0YlRi8=

factor(subs(x=0,algLxt0Zm));degree(%,Lxt0);

LCQqLCIkYyMiIiIpLCYqJClJI1ptRzYiIiIjRiVGJSIiJEYlRi1GJSlGKiIiJ0YlSSVMeHQwR0YrRiUsLComRjBGJUYuRiVGJSooIiIqRiVGMEYlKUYqIiIlRiVGJSooIiNGRiVGMEYlRilGJUYlKiYiI2tGJSlGKkYtRiUhIiIqJkY4RiVGMEYlRiVGJUYl

IiIj

factor([solve(%%,Lxt0)]);

NyQiIiEsJCooIiNrIiIiKUkjWm1HNiIiIiRGJyksJiokKUYpIiIjRidGJ0YrRidGKyEiIkYn

L00Zm;

LCQqKCIjayIiIilJI1ptRzYiIiIkRiUpLCYqJClGJyIiI0YlRiVGKUYlRikhIiJGJQ==

factor(subs(Lxt0=L00Zm+x*Lhat,algLxt0Zm))/x: degree(subs(x=0,%),Lhat);

IiIi

Is the coefficient of x^(2i-1) in L(xt,0) a fraction in T=Z^2 ? no

map(factor,op(1,algeqtoseries(algLxt0Zm,x,Lxt0,4,true)));

Ky1JInhHNiIsJCooIiNrIiIiKUkjWm1HRiQiIiRGKCksJiokKUYqIiIjRihGKEYrRihGKyEiIkYoIiIhLCQqKiIjO0YoLCZGKkYoRihGMUYoLCZGKkYoRihGKEYoRixGMUYoRigsJCouRjBGKEY3RigsKiokRilGKEYoKiZGK0YoRi9GKEYxRipGMUYoRjFGKClGNkYwRihGKUYxRixGMUYxRjAsJCouI0YoRjVGKEY3RigsMComRitGKClGKiIiJ0YoRigqJiIjRUYoKUYqIiImRihGMSomIiNMRigpRioiIiVGKEYoKiYiI0dGKEYpRihGKComIiNIRihGL0YoRigqJkYwRihGKkYoRjFGKEYxRigpRjZGK0YoRkNGMUYsRjFGKEYrLUkiT0clKnByb3RlY3RlZEc2I0YoRkw=

expand(numer(coeff(%,x,3)));

LDYqJiIiJCIiIilJI1ptRzYiIiM1RiVGJSomIiNLRiUpRiciIipGJSEiIiomIiMmKUYlKUYnIiIpRiVGJSomRitGJSlGJyIiKEYlRi4qJiIjIylGJSlGJyIiJ0YlRi4qJkYrRiUpRiciIiZGJUYlKiYiI0VGJSlGJyIiJUYlRiUqJkYrRiUpRidGJEYlRiUqJiIjTEYlKUYnIiIjRiVGLkYlRiU=

Now for L(0,xt)

algL0xtZm;

degree 1 in L0xt

factor(subs(x=0,algL0xtZm));degree(%,L0xt);

LCQqKCIrJ0huXEglIiIiKUkjWm1HNiIiI0NGJSwsKiZJJUwweHRHRihGJSlGJyIiJ0YlRiUqKCIiKkYlRixGJSlGJyIiJUYlRiUqKCIjRkYlRixGJSlGJyIiI0YlRiUqJiIja0YlKUYnIiIkRiUhIiIqJkY0RiVGLEYlRiVGJUY7

IiIi

factor([solve(%%,L0xt)]);

NyMsJCooIiNrIiIiKUkjWm1HNiIiIiQiIiIpLCYqJClJI1ptRzYiIiIjIiIiIiIiIiIkIiIiIiIkISIiIiIi

Now for B(xt,0)

algBxt0Zm;

series(subs(x=x*t,y=0,Bser(4)),t,5);

KylJInRHNiIiIiIiIiMsJiIjNUYlKiZGJkYlSSJ4R0YkRiVGJSIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

degree 1 again

factor(subs(x=0,algBxt0Zm));degree(%,Bxt0);

LCQqKCIrI2ZNKiplKSIiIilJI1ptRzYiIiNDRiUsOiomSSVCeHQwR0YoRiUpRiciIihGJUYlKiZGLEYlKUYnIiInRiVGJSooIiIqRiVGLEYlKUYnIiImRiVGJSomIiNLRiVGMEYlRiUqKEYzRiVGLEYlKUYnIiIlRiVGJSomIiMnKkYlRjRGJSEiIiooIiNGRiVGLEYlKUYnIiIkRiVGJSomRjdGJUY5RiVGJSooRj9GJUYsRiUpRiciIiNGJUYlKiZGN0YlRkBGJUYlKihGP0YlRixGJUYnRiVGJSomRj9GJUYsRiVGJUYlRiU=

IiIi

factor([solve(%%,Bxt0)]);

NyMsJCouIiNLIiIiLCZJI1ptRzYiIiIiIiIiISIiIiIiLCgqJClJI1ptRzYiIiIjIiIiIiIiKiYiIiMiIiJJI1ptRzYiIiIiISIiIiIiISIiIiIiKUkjWm1HNiIiIiQiIiIsJkkjWm1HNiIiIiIiIiIiIiIhIiIpLCYqJClJI1ptRzYiIiIjIiIiIiIiIiIkIiIiIiIkISIiISIi

B00Zm;

LCQqLiIjSyIiIiwmSSNabUc2IkYlRiUhIiJGJSwoKiQpRiciIiNGJUYlKiZGLUYlRidGJUYpRiVGKUYlKUYnIiIkRiUsJkYnRiVGJUYlRikpLCZGK0YlRjBGJUYwRilGKQ==

Now for B(0,xt)

algB0xtZm;

factor(subs(x=0,algB0xtZm));degree(%,B0xt);

LCQqKCImT2InIiIiKUkjWm1HNiIiIzdGJSw6KiZJJUIweHRHRihGJSlGJyIiKEYlRiUqJkYsRiUpRiciIidGJUYlKigiIipGJUYsRiUpRiciIiZGJUYlKiYiI0tGJUYwRiVGJSooRjNGJUYsRiUpRiciIiVGJUYlKiYiIycqRiVGNEYlISIiKigiI0ZGJUYsRiUpRiciIiRGJUYlKiZGN0YlRjlGJUYlKihGP0YlRixGJSlGJyIiI0YlRiUqJkY3RiVGQEYlRiUqKEY/RiVGLEYlRidGJUYlKiZGP0YlRixGJUYlRiVGJQ==

IiIi

factor([solve(%%,B0xt)]);

We can now conclude that the series Cij belongs to t^(i+j-1) Q(T). Let us compute some of these series explicitly

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,0): the series L00</Text-field>

L00Zm;

LCQqKCIjayIiIilJI1ptRzYiIiIkRiUpLCYqJClGJyIiI0YlRiVGKUYlRikhIiJGJQ==

Cm10sol:=L00Zm;

LCQqKCIjayIiIilJI1ptRzYiIiIkRiUpLCYqJClGJyIiI0YlRiVGKUYlRikhIiJGJQ==

n:=10: normal(series(subs(Zm=Zmser(n),Cm10sol),t,n+1));

Ky9JInRHNiIiIiIiIiEiIiQiIiMiI0AiIiUiJCQ+IiInIiVNPyIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

coeff(coeff(convert(Cser(n),polynom),x,-1),y,0);

LC4qJiImJEdCIiIiKUkidEc2IiIjNUYlRiUqJiIlTT9GJSlGJyIiKUYlRiUqJiIkJD5GJSlGJyIiJ0YlRiUqJiIjQEYlKUYnIiIlRiVGJSomIiIkRiUpRiciIiNGJUYlRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,1) and (-1,2) and (-1,3): the series L01, L02 and L03</Text-field>

with(gfun): map(factor,op(algeqtoseries(algL0xtZm,x,L0xt,4,true)));

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

L01ftZm:=coeff(%,x); L02Zm:=factor(coeff(%%,x,2)/t2Zm); L03ftZm:=factor(coeff(%%%,x,3)/t2Zm);

LCQqLCIjOyIiIilJI1ptRzYiIiIjRiUsJkYnRiVGJSEiIkYlLCZGJ0YlIiIkRitGJSksJiokRiZGJUYlRi1GJUYtRitGKw==

LCQqLiIjayIiIilJI1ptRzYiIiIkRiUsJkYnRiVGJSEiIkYlLDAqJiIiKkYlKUYnIiInRiVGJSomIiN5RiUpRiciIiZGJUYrKiYiJEIiRiUpRiciIiVGJUYlKiYiI09GJUYmRiVGJSomIiIoRiUpRiciIiNGJUYlKiYiI0FGJUYnRiVGKyIjNkYrRiUpLCYqJEY9RiVGJUYpRiVGMEYrLCZGJ0YlRiVGJUYrRiU=

LCQqLiIjayIiIilJI1ptRzYiIiIjRiUpLCZGJ0YlRiUhIiJGKUYlLDIqJClGJyIiKEYlRiUqJiIjPEYlKUYnIiInRiVGLComIiN0RiUpRiciIiZGJUYlKiYiI2xGJSlGJyIiJUYlRiwqJiIjXEYlKUYnIiIkRiVGLComIiNKRiVGJkYlRiwqJiIjQkYlRidGJUYlIiNMRiVGJSksJiokRiZGJUYlRkBGJUY0RiwsJkYnRiVGJUYlRixGLA==

coeff(coeff(Lser(n),x,0),y,1);

LCwqJiIlRCcqIiIiKUkidEc2IiIiKkYlRiUqJiIkSSlGJSlGJyIiKEYlRiUqJiIjeEYlKUYnIiImRiVGJSomIiIpRiUpRiciIiRGJUYlRidGJQ==

n:=12:normal(series(subs(Zm=Zmser(n),L01ftZm),t,n+1));

Ky9JInRHNiIiIiIiIiMiIikiIiUiI3giIiciJEkpRiciJUQnKiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

Cm11ftsol:=L01ftZm; Cm12sol:=L02Zm; Cm13ftsol:=L03ftZm;

LCQqLCIjOyIiIilJI1ptRzYiIiIjRiUsJkYnRiVGJSEiIkYlLCZGJ0YlIiIkRitGJSksJiokRiZGJUYlRi1GJUYtRitGKw==

n:=12:normal(series(subs(Zm=Zmser(n),Cm11ftsol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIikiIiUiI3giIiciJEkpRiciJUQnKiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

coeff(coeff(convert(Cser(n),polynom),x,-1),y,1);

LC4qJiInNXU2IiIiKUkidEc2IiIjNkYlRiUqJiIlRCcqRiUpRiciIipGJUYlKiYiJEkpRiUpRiciIihGJUYlKiYiI3hGJSlGJyIiJkYlRiUqJiIiKUYlKUYnIiIkRiVGJUYnRiU=

n:=12:normal(series(subs(Zm=Zmser(n),Cm12sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIzoiIiUiJCM+IiInIiVSQyIiKSImdzkkIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

coeff(coeff(convert(Cser(n),polynom),x,-1),y,2);

LC4qJiInTVJUIiIiKUkidEc2IiIjN0YlRiUqJiImdzkkRiUpRiciIzVGJUYlKiYiJVJDRiUpRiciIilGJUYlKiYiJCM+RiUpRiciIidGJUYlKiYiIzpGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

n:=12:normal(series(subs(Zm=Zmser(n),Cm13ftsol),t,n+1));

Ky9JInRHNiIiIiIiIiUiI0MiIiciJCdSIiIpIiUvZiIjNSImKjMmKSIjNy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,-1),y,3);

LCwqJiImKjMmKSIiIilJInRHNiIiIzZGJUYlKiYiJS9mRiUpRiciIipGJUYlKiYiJCdSRiUpRiciIihGJUYlKiYiI0NGJSlGJyIiJkYlRiUqJClGJyIiJEYlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0): the series L10 (in Q(T))</Text-field>

map(factor,op(1,algeqtoseries(algLxt0Zm,x,Lxt0,2,true)));

KylJInhHNiIsJCooIiNrIiIiKUkjWm1HRiQiIiRGKCksJiokKUYqIiIjRihGKEYrRihGKyEiIkYoIiIhLCQqKiIjO0YoLCZGKkYoRihGMUYoLCZGKkYoRihGKEYoRixGMUYoRigtSSJPRyUqcHJvdGVjdGVkRzYjRihGMA==

L10ftZm:=coeff(%,x);

LCQqKiIjOyIiIiwmSSNabUc2IkYlRiUhIiJGJSwmRidGJUYlRiVGJSksJiokKUYnIiIjRiVGJSIiJEYlRjBGKUYl

Cm20ftsol:=L10ftZm:

coeff(coeff(convert(Cser(n),polynom),x,-2),y,0);

LC4qJiImd28nIiIiKUkidEc2IiIjNkYlRiUqJiIlbmNGJSlGJyIiKkYlRiUqJiIkNyZGJSlGJyIiKEYlRiUqJiIjXkYlKUYnIiImRiVGJSomIiInRiUpRiciIiRGJUYlRidGJQ==

n:=16: normal(series(subs(Zm=Zmser(n),Cm20ftsol),t,n+1));

KzNJInRHNiIiIiIiIiMiIiciIiUiI15GJyIkNyYiIikiJW5jIiM1IiZ3byciIzciJ15kIykiIzktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzs=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,0) and (-4,0): the series L20 and L30</Text-field>

map(factor,op(1,algeqtoseries(algLxt0Zm,x,Lxt0,4,true)));

L20Zm:=factor(coeff(%,x,2)/t2Zm); L30ftZm:=factor(coeff(%%,x,3)/t2Zm);

LCQqLCIkNyYiIiIpSSNabUc2IiIiJEYlLCZGJ0YlRiUhIiJGJSwqKiRGJkYlRiUqJkYpRiUpRiciIiNGJUYrRidGK0YlRitGJSksJiokRi9GJUYlRilGJSIiJ0YrRis=

LCQqKiIjOyIiIiksJkkjWm1HNiJGJUYlISIiIiIjRiUsMComIiIkRiUpRigiIidGJUYlKiYiI0VGJSlGKCIiJkYlRioqJiIjTEYlKUYoIiIlRiVGJSomIiNHRiUpRihGLkYlRiUqJiIjSEYlKUYoRitGJUYlKiZGK0YlRihGJUYqRiVGKkYlKSwmKiRGPkYlRiVGLkYlRjBGKkYl

Cm30sol:=L20Zm: Cm40ftsol:=L30ftZm:

coeff(coeff(convert(Cser(n),polynom),x,-3),y,0);

LDIqJiIpbCcqPkkiIiIpSSJ0RzYiIiM7RiVGJSomIihNdEgjRiUpRiciIzlGJUYlKiYiJ1gheiJGJSlGJyIjN0YlRiUqJiImKVE5RiUpRiciIzVGJUYlKiYiJS43RiUpRiciIilGJUYlKiYiJDEiRiUpRiciIidGJUYlKiZGNUYlKUYnIiIlRiVGJSokKUYnIiIjRiVGJQ==

n:=16: normal(series(subs(Zm=Zmser(n),Cm30sol),t,n+1));

KzNJInRHNiIiIiIiIiMiIzUiIiUiJDEiIiInIiUuNyIiKSImKVE5RiciJ1gheiIiIzciKE10SCMiIzktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzs=

coeff(coeff(convert(Cser(n),polynom),x,-4),y,0);

LDAqJiIoYCJ6ZiIiIilJInRHNiIiIzpGJUYlKiYiJy5QV0YlKUYnIiM4RiVGJSomIiY3TSRGJSlGJyIjNkYlRiUqJiIlYERGJSlGJyIiKkYlRiUqJiIkKD5GJSlGJyIiKEYlRiUqJkYpRiUpRiciIiZGJUYlKiQpRiciIiRGJUYl

n:=16: normal(series(subs(Zm=Zmser(n),Cm40ftsol),t,n+1));

KzNJInRHNiIiIiIiIiUiIzoiIiciJCg+IiIpIiVgRCIjNSImN00kIiM3IicuUFciIzkiKGAiemYiIzstSSJPRyUqcHJvdGVjdGVkRzYjRiUiIz0=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,1): the series L11</Text-field>

collect(convert(series(convert(series(subs(Ker=K,eqL),x,3),polynom),y,2),polynom),[x,y],distributed,factor);

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

expand(solve(coeff(coeff(%,x,1),y,0),(D[1, 2](L))(0, 0)));

LCgtSSJMRzYiNiQiIiFGJyEiIiomIyIiIiIiI0YrLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRGK0YrNiNGJEYmRitGKComSSJ0R0YlRigtLSZGMDYjRitGNUYmRitGKw==

L11Zm:=factor(-L00Zm-L20Zm+L10ftZm/t2Zm);

LCQqLCIjayIiIiwmSSNabUc2IkYlRiUhIiJGJSwuKiQpRiciIiZGJUYlKiQpRiciIiVGJUYlKiYiIiNGJSlGJyIiJEYlRiUqJiIjSUYlKUYnRjJGJUYpKiYiIz5GJUYnRiVGKUY5RilGJUYzRiUpLCYqJEY3RiVGJUY0RiUiIidGKUYp

Cm21sol:=L11Zm;

LCQqLCIjayIiIiwmSSNabUc2IkYlRiUhIiJGJSwuKiQpRiciIiZGJUYlKiQpRiciIiVGJUYlKiYiIiNGJSlGJyIiJEYlRiUqJiIjSUYlKUYnRjJGJUYpKiYiIz5GJUYnRiVGKUY5RilGJUYzRiUpLCYqJEY3RiVGJUY0RiUiIidGKUYp

n:=12: series(subs(Zm=Zmser(n),Cm21sol),t,n+1);

Ky9JInRHNiIiIiNGJSIjPyIiJSIkOCMiIiciJUlDIiIpIiYwI0giIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

coeff(coeff(convert(Cser(n),polynom),x,-2),y,1);

LC4qJiInPV1PIiIiKUkidEc2IiIjN0YlRiUqJiImMCNIRiUpRiciIzVGJUYlKiYiJUlDRiUpRiciIilGJUYlKiYiJDgjRiUpRiciIidGJUYlKiYiIz9GJSlGJyIiJUYlRiUqJiIiI0YlKUYnRjtGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-1) and (0,-2) and (0,-3) and (0,-4): the series B00, B01 (in Q(T)), B02, B03</Text-field>

ser:=map(factor,op(algeqtoseries(algB0xtZm,x,B0xt,4,true)));

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

C0m1sol:=coeff(%,x,0); C0m2ftsol:=coeff(%%,x,1); C0m3sol:=factor(coeff(%%%,x,2)/t2Zm); C0m4ftsol:=factor(coeff(ser,x,3)/t2Zm);

LCQqLiIjSyIiIiwmSSNabUc2IkYlRiUhIiJGJSwoKiQpRiciIiNGJUYlKiZGLUYlRidGJUYpRiVGKUYlKUYnIiIkRiUsJkYnRiVGJUYlRikpLCZGK0YlRjBGJUYwRilGKQ==

LCQqKiwmKiQpSSNabUc2IiIiIyIiIkYqIiImISIiRiopLCZGJ0YqRipGLEYpRiopLCZGJ0YqRipGKkYpRiopLCZGJUYqIiIkRipGM0YsRiw=

LCQqLiIkNyYiIiIpSSNabUc2IiIiJkYlKSwmRidGJUYlISIiIiIjRiUsKiokKUYnIiIkRiVGJSokKUYnRi1GJUYsRidGLEYxRixGJSksJkYyRiVGMUYlIiInRiwsJkYnRiVGJUYlRixGLA==

LCQqLCIjOyIiIiksJkkjWm1HNiJGJUYlISIiIiIkRiUsNComRitGJSlGKCIiKUYlRiUqJiIjP0YlKUYoIiIoRiVGKiomIiIqRiUpRigiIidGJUYlKiZGL0YlKUYoIiImRiVGJSomIiNmRiUpRigiIiVGJUYlKiZGMUYlKUYoRitGJUYlKiZGOkYlKUYoIiIjRiVGKiomRi9GJUYoRiVGKkZDRipGJSksJiokRkJGJUYlRitGJUY3RiosJkYoRiVGJUYlRipGJQ==

B00Zm:=C0m1sol; B01ftZm:=C0m2ftsol; B02Zm:=C0m3sol; B03ftZm:=C0m4ftsol;

LCQqLiIjSyIiIiwmSSNabUc2IkYlRiUhIiJGJSwoKiQpRiciIiNGJUYlKiZGLUYlRidGJUYpRiVGKUYlKUYnIiIkRiUsJkYnRiVGJUYlRikpLCZGK0YlRjBGJUYwRilGKQ==

LCQqKiwmKiQpSSNabUc2IiIiIyIiIkYqIiImISIiRiopLCZGJ0YqRipGLEYpRiopLCZGJ0YqRipGKkYpRiopLCZGJUYqIiIkRipGM0YsRiw=

LCQqLiIkNyYiIiIpSSNabUc2IiIiJkYlKSwmRidGJUYlISIiIiIjRiUsKiokKUYnIiIkRiVGJSokKUYnRi1GJUYsRidGLEYxRixGJSksJkYyRiVGMUYlIiInRiwsJkYnRiVGJUYlRixGLA==

n:=12: normal(series(subs(Zm=Zmser(n),C0m1sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIzUiIiUiJDMiIiInIiVfNyIiKSImel8iRictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

coeff(coeff(convert(Cser(n),polynom),x,0),y,-1);

LC4qJiInV08+IiIiKUkidEc2IiIjN0YlRiUqJiImel8iRiUpRiciIzVGJUYlKiYiJV83RiUpRiciIilGJUYlKiYiJDMiRiUpRiciIidGJUYlKiZGLUYlKUYnIiIlRiVGJSokKUYnIiIjRiVGJQ==

n:=12: normal(series(subs(Zm=Zmser(n),C0m2ftsol),t,n+1));

Ky9JInRHNiIiIiIiIiUiIzkiIiciJCE9IiIpIiVDQiIjNSImRTAkIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlRic=

coeff(coeff(convert(Cser(n),polynom),x,0),y,-2);

LCwqJiImRTAkIiIiKUkidEc2IiIjNkYlRiUqJiIlQ0JGJSlGJyIiKkYlRiUqJiIkIT1GJSlGJyIiKEYlRiUqJiIjOUYlKUYnIiImRiVGJSokKUYnIiIkRiVGJQ==

n:=12: normal(series(subs(Zm=Zmser(n),C0m3sol),t,n+1));

Ky9JInRHNiIiIiIiIiUiIz4iIiciJCNIIiIpIiVUVSIjNSImdDAnIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlIiM5

coeff(coeff(convert(Cser(n),polynom),x,0),y,-3);

LCwqJiImdDAnIiIiKUkidEc2IiIjN0YlRiUqJiIlVFVGJSlGJyIjNUYlRiUqJiIkI0hGJSlGJyIiKUYlRiUqJiIjPkYlKUYnIiInRiVGJSokKUYnIiIlRiVGJQ==

n:=12: normal(series(subs(Zm=Zmser(n),C0m4ftsol),t,n+1));

Ky1JInRHNiIiIiIiIiciI0QiIikiJGQlIiM1IiUmWygiIzctSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzk=

coeff(coeff(convert(Cser(n),polynom),x,0),y,-4);

LCoqJiIlJlsoIiIiKUkidEc2IiIjNkYlRiUqJiIkZCVGJSlGJyIiKkYlRiUqJiIjREYlKUYnIiIoRiVGJSokKUYnIiImRiVGJQ==

B02Zm := -512*Zm^5*(Zm-1)^2*(Zm^3-Zm^2-Zm-3)/((Zm^2+3)^6*(Zm+1));

LCQqLiIkNyYiIiIpSSNabUc2IiIiJkYlKSwmRidGJUYlISIiIiIjRiUsKiokKUYnIiIkRiVGJSokKUYnRi1GJUYsRidGLEYxRixGJSksJkYyRiVGMUYlIiInRiwsJkYnRiVGJUYlRixGLA==

B03ftZm := 16*(Zm-1)^3*(3*Zm^8-20*Zm^7+9*Zm^6+8*Zm^5+59*Zm^4+20*Zm^3-5*Zm^2-8*Zm-2)/((Zm^2+3)^6*(Zm+1));

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,2): the series L12</Text-field>

collect(convert(series(convert(series(subs(Ker=K,eqL),x,4),polynom),y,4),polynom),[x,y],distributed);

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

expand(solve(coeff(coeff(%,x,0),y,2),(D[1, 2,2](L))(0, 0)));

LCotLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIzYjSSJMR0YqNiQiIiFGMCEiIiomIyIiIiIiJ0Y0LS0mRiY2JUYsRixGLEYtRi9GNEYxKiZGM0Y0LS1GODYjSSJCR0YqRi9GNEYxKigjRjRGLEY0SSJ0R0YqRjEtLSZGJjYkRixGLEYtRi9GNEY0

L12ftZm:=factor((-L03ftZm-L01ftZm-B03ftZm+L02Zm)/2);

LCQqKiIjOyIiIiksJkkjWm1HNiJGJUYlISIiIiIjRiUsNCokKUYoIiIpRiVGJSomIiInRiUpRigiIihGJUYqKiZGK0YlKUYoRjFGJUYlKiYiI0lGJSlGKCIiJkYlRioqJiIjISlGJSlGKCIiJUYlRiUqJiIjUUYlKUYoIiIkRiVGJSomIiQ1IkYlKUYoRitGJUYlKiZGK0YlRihGJUYqRiVGKkYlKSwmKiRGREYlRiVGQUYlRjFGKkYl

Cm22ftsol:=%;

n:=12: series(subs(Zm=Zmser(n),Cm22ftsol),t,n+1);

Ky9JInRHNiIiIiQiIiUiI1giIiciJCVmIiIpIiVYeCIjNSIndj41IiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,-2),y,2);

LCwqJiIndj41IiIiKUkidEc2IiIjNkYlRiUqJiIlWHhGJSlGJyIiKkYlRiUqJiIkJWZGJSlGJyIiKEYlRiUqJiIjWEYlKUYnIiImRiVGJSomIiIkRiUpRidGN0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,1): the series L21 (in Q(T))</Text-field>

collect(convert(series(convert(series(subs(Ker=K,eqL),x,3),polynom),y,2),polynom),[x,y],distributed);

expand(solve(coeff(coeff(%,x,2),y,0),(D[1, 1,2](L))(0, 0)));

LCgqJiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiU2I0kiTEdGLTYkIiIhRjJGJSEiIiomI0YlIiIkRiUtLSZGKTYlRiVGJUYlRi9GMUYlRjMqJkkidEdGLUYzLS0mRik2JEYlRiVGL0YxRiVGJQ==

L21ftZm:=factor(-L10ftZm-L30ftZm+L20Zm);

LCQqLiIjayIiIiksJkkjWm1HNiJGJUYlISIiIiIjRiUsJiokKUYoRitGJUYlIiIoRipGJSwmRi1GJUYlRiVGJSksJkYoRiVGJUYlRitGJSksJkYtRiUiIiRGJSIiJ0YqRio=

Cm31ftsol:=%;

LCQqLiIjayIiIiksJkkjWm1HNiJGJUYlISIiIiIjRiUsJiokKUYoRitGJUYlIiIoRipGJSwmRi1GJUYlRiVGJSksJkYoRiVGJUYlRitGJSksJkYtRiUiIiRGJSIiJ0YqRio=

n:=12: series(subs(Zm=Zmser(n),Cm31ftsol),t,n+1);

Ky9JInRHNiIiIiQiIiUiI1MiIiciJCVcIiIpIiVvaCIjNSImZCh5IiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,-3),y,1);

LCwqJiImZCh5IiIiKUkidEc2IiIjNkYlRiUqJiIlb2hGJSlGJyIiKkYlRiUqJiIkJVxGJSlGJyIiKEYlRiUqJiIjU0YlKUYnIiImRiVGJSomIiIkRiUpRidGN0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-1) and (2,-1) and (3,-1): the series B10, B20 B30</Text-field>

map(factor,op(algeqtoseries(algBxt0Zm,x,Bxt0,4,true)));

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

C1m1ftsol:=coeff(%,x,1); C2m1sol:=factor(coeff(%%,x,2)/t2Zm); C3m1ftsol:=factor(coeff(%%%,x,3)/t2Zm);

LCQqLCIiIyIiIiwuKiQpSSNabUc2IiIiJkYlRiUqJiIjOEYlKUYpIiIlRiUhIiIqJiIjOUYlKUYpIiIkRiVGJSomIiM9RiUpRilGJEYlRiUqJiIiKkYlRilGJUYlRjRGJUYlKSwmRilGJUYlRjBGJEYlLCZGKUYlRiVGJUYwKSwmKiRGN0YlRiVGNEYlRjRGMEYl

LCQqLiIjSyIiIilJI1ptRzYiIiIkRiUpLCZGJ0YlRiUhIiIiIiNGJSw0KiQpRiciIilGJUYlKiZGJEYlKUYnIiIoRiVGLComIiRtIkYlKUYnIiInRiVGJSomIiNvRiUpRiciIiZGJUYsKiYiJDcjRiUpRiciIiVGJUYsKiYiJCsjRiVGJkYlRiwqJiIjJSpGJSlGJ0YtRiVGLComIiNXRiVGJ0YlRiUiIzZGJUYlKSwmRidGJUYlRiVGLUYsKSwmKiRGRUYlRiVGKUYlRjhGLEYs

LCQqLCIiIyIiIiksJkkjWm1HNiJGJUYlISIiIiIkRiUsOiokKUYoIiM2RiVGJSomIiNmRiUpRigiIzVGJUYqKiYiJHInRiUpRigiIipGJUYlKiYiJUw+RiUpRigiIilGJUYqKiYiJGkjRiUpRigiIihGJUYqKiYiJUE+RiUpRigiIidGJUYlKiYiJTlLRiUpRigiIiZGJUYlKiYiJT1ERiUpRigiIiVGJUYlKiYiJHooRiUpRihGK0YlRioqJiIkYilGJSlGKEYkRiVGKiomIiQmR0YlRihGJUYqIiNkRipGJSksJkYoRiVGJUYlRiRGKiksJiokRlFGJUYlRitGJUZDRipGJQ==

B10ftZm:=C1m1ftsol; B20Zm:=C2m1sol; B30ftZm:=C3m1ftsol;

LCQqLCIiIyIiIiwuKiQpSSNabUc2IiIiJkYlRiUqJiIjOEYlKUYpIiIlRiUhIiIqJiIjOUYlKUYpIiIkRiVGJSomIiM9RiUpRilGJEYlRiUqJiIiKkYlRilGJUYlRjRGJUYlKSwmRilGJUYlRjBGJEYlLCZGKUYlRiVGJUYwKSwmKiRGN0YlRiVGNEYlRjRGMEYl

n:=12: normal(series(subs(Zm=Zmser(n),C1m1ftsol),t,n+1));

Ky9JInRHNiIiIiMiIiUiI0giIiciJCFRIiIpIiVrXCIjNSImO2QnIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,1),y,-1);

LCwqJiImO2QnIiIiKUkidEc2IiIjNkYlRiUqJiIla1xGJSlGJyIiKkYlRiUqJiIkIVFGJSlGJyIiKEYlRiUqJiIjSEYlKUYnIiImRiVGJSomIiIjRiUpRiciIiRGJUYl

n:=12: normal(series(subs(Zm=Zmser(n),C2m1sol),t,n+1));

Ky9JInRHNiIiIiQiIiUiI2ciIiciJFwqIiIpIiZQUyIiIzUiJzJJPyIjNy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzk=

coeff(coeff(convert(Cser(n),polynom),x,2),y,-1);

LCwqJiInMkk/IiIiKUkidEc2IiIjN0YlRiUqJiImUFMiRiUpRiciIzVGJUYlKiYiJFwqRiUpRiciIilGJUYlKiYiI2dGJSlGJyIiJ0YlRiUqJiIiJEYlKUYnIiIlRiVGJQ==

n:=12: normal(series(subs(Zm=Zmser(n),C3m1ftsol),t,n+1));

Ky1JInRHNiIiIiUiIiciJDEiIiIpIiUvPyIjNSImXE4kIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,3),y,-1);

LCoqJiImXE4kIiIiKUkidEc2IiIjNkYlRiUqJiIlLz9GJSlGJyIiKkYlRiUqJiIkMSJGJSlGJyIiKEYlRiUqJiIiJUYlKUYnIiImRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-2): the series B11</Text-field>

collect(map(Z->convert(series(Z,y,3),polynom),convert(series(subs(Ker=K,eqB),x,4),polynom)),[x,y],distributed,factor);

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

coeff(coeff(%,x,1),y,0);

LCwqKCMiIiIiIiNGJUkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiRGJUYlNiNJIkJHRig2JCIiIUY0RiVGJSooRiZGJUYnRiUtLSZGLDYkRiVGJkYxRjNGJUYlLS0mRiw2I0YlRjFGMyEiIiomRidGJS1GMkYzRiVGJSooRiRGJUYnRiUtLUYrNiNJIkxHRihGM0YlRiU=

expand(isolate(%,(D[1, 2](B))(0, 0)));

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkIiIiIiIjNiNJIkJHRio2JCIiIUYxLCoqJiNGLCIiJUYsLS0mRiY2JEYsRixGLkYwRiwhIiIqKCNGLEYtRixJInRHRipGOi0tJkYmNiNGLEYuRjBGLEYsKiZGPEYsLUYvRjBGLEY6KiZGNEYsLS1GODYjSSJMR0YqRjBGLEY6

B11Zm:=factor(-1/2*L20Zm-1/2*B20Zm+1/2*B10ftZm/t2Zm-1/2*B00Zm);

LCQqLiIjSyIiIilJI1ptRzYiIiIkRiUpLCZGJ0YlRiUhIiIiIiNGJSwyKiQpRiciIihGJUYlKiYiIipGJSlGJyIiJ0YlRiwqJClGJyIiJkYlRiwqJiIjPkYlKUYnIiIlRiVGLComIiN6RiVGJkYlRiUqJiIjTEYlKUYnRi1GJUYlKiYiIyIpRiVGJ0YlRiUiI0ZGJUYlKSwmKiRGQUYlRiVGKUYlRjVGLCwmRidGJUYlRiVGLEYl

C1m2sol:=B11Zm:

n:=12: normal(series(subs(Zm=Zmser(n),C1m2sol),t,n+1));

Ky9JInRHNiIiIiQiIiUiI2AiIiciJCF5IiIpIiYxNSIiIzUiJ3BSOiIjNy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzk=

coeff(coeff(convert(Cser(n),polynom),x,1),y,-2);

LCwqJiIncFI6IiIiKUkidEc2IiIjN0YlRiUqJiImMTUiRiUpRiciIzVGJUYlKiYiJCF5RiUpRiciIilGJUYlKiYiI2BGJSlGJyIiJ0YlRiUqJiIiJEYlKUYnIiIlRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (2,-2): the series B21 </Text-field>

collect(map(Z->convert(series(Z,y,4),polynom),convert(series(subs(Ker=K,eqB),x,4),polynom)),[x,y],distributed,factor);

LDwqJkkidEc2IiIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiY2I0kiQkdGJTYkIiIhRjJGJkYmKiZGJEYmLS1GKTYjSSJMR0YlRjFGJkYmKigiIiNGJkYkRiYtLSZGKjYjRjlGL0YxRiZGJi1GMEYxISIiKiojRiYiIiRGJkYkRiYtLSZGKjYmRiZGOUY5RjlGL0YxRiYpSSJ5R0YlRkJGJkYmKigsLComRiRGJi0tJkYqNiVGJkYmRjlGL0YxRiZGJiomRiRGJi0tJkYqNiVGJkY5RjlGL0YxRiZGJiomRjlGJi0tJkYqNiRGJkY5Ri9GMUYmRj8qKEY5RiZGJEYmRidGJkYmRjhGJkYmSSJ4R0YlRiZGSEYmRiYqKCwsKihGQUYmRiRGJi0tJkYqNiZGJkYmRiZGOUYvRjFGJkYmKigjRiZGOUYmRiRGJi0tJkYqNiZGJkYmRjlGOUYvRjFGJkYmRkxGPyomRiRGJi0tJkYqNiRGJkYmRi9GMUYmRiYqKEY5RiZGJEYmRlZGJkYmRiYpRmVuRjlGJkZIRiZGJiomLCpGUEYmKihGQUYmRiRGJi0tJkYqNiVGOUY5RjlGL0YxRiZGJi0tJkYqNiRGOUY5Ri9GMUY/RjhGJkYmKUZIRjlGJkYmKigsLEZdb0YmKihGQUYmRiRGJkZDRiZGJkZRRj9GaG9GJiomRiRGJkZhcEYmRiZGJkZlbkYmRmVwRiZGJiooLCwqKCNGJiIiJ0YmRiRGJi0tJkYqNidGJkYmRiZGOUY5Ri9GMUYmRiYqKEZdcUYmRiRGJi0tJkYqNidGJkYmRjlGOUY5Ri9GMUYmRiYqJkZeb0YmRl9vRiZGP0ZLRiZGUEYmRiZGaW9GJkZlcEYmRiYqJiwsKihGXm9GJkYkRiZGZG9GJkYmRmhvRiZGJ0Y/KiZGJEYmRj5GJkYmKihGXm9GJkYkRiYtLUZmb0Y2RjFGJkYmRiZGZW5GJkYmKiYsKkZpcEYmRmhvRiYqKEY5RiZGJEYmRj5GJkYmKiZGOUYmRjpGJkY/RiZGSEYmRiYqJiwsKihGXXFGJkYkRiYtLSZGKjYlRiZGJkYmRi9GMUYmRiZGS0YmKiZGXm9GJkZkb0YmRj9GI0YmKihGXXFGJkYkRiYtLUZpckY2RjFGJkYmRiZGaW9GJkYm

coeff(coeff(%,x,2),y,0);

LCwqKCMiIiIiIidGJUkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiVGJUYlRiU2I0kiQkdGKDYkIiIhRjRGJUYlKiZGJ0YlLS0mRiw2JUYlRiUiIiNGMUYzRiVGJSomI0YlRjpGJS0tJkYsNiRGJUYlRjFGM0YlISIiKiZGJ0YlLS0mRiw2I0YlRjFGM0YlRiUqKEYkRiVGJ0YlLS1GKzYjSSJMR0YoRjNGJUYl

expand(isolate(%,(D[1, 1,2](B))(0, 0)));

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlIiIiRiwiIiM2I0kiQkdGKjYkIiIhRjEsKiomI0YsIiInRiwtLSZGJjYlRixGLEYsRi5GMEYsISIiKigjRixGLUYsSSJ0R0YqRjotLSZGJjYkRixGLEYuRjBGLEYsLS0mRiY2I0YsRi5GMEY6KiZGNEYsLS1GODYjSSJMR0YqRjBGLEY6

B21ftZm:=factor((-L30ftZm-B30ftZm+B20Zm-B10ftZm)/2);

LCQqLCIiIyIiIiksJkkjWm1HNiJGJUYlISIiIiIkRiUsOCokKUYoIiM1RiVGJSomIiNHRiUpRigiIipGJUYqKiYiJDoiRiUpRigiIilGJUYlKiYiI1NGJSlGKCIiKEYlRioqJiIkXSVGJSlGKCIiJ0YlRiUqJiIkTyhGJSlGKCIiJkYlRioqJiIkPSVGJSlGKCIiJUYlRioqJiIlUzdGJSlGKEYrRiVGKiomIiReKUYlKUYoRiRGJUYqKiYiJGcjRiVGKEYlRioiI2xGKkYlKSwmKiRGTUYlRiVGK0YlRj9GKiwmRihGJUYlRiVGKkYq

C2m2ftsol:=B21ftZm:

n:=12: normal(series(subs(Zm=Zmser(n),C2m2ftsol),t,n+1));

Ky1JInRHNiIiIidGJSIkTCIiIikiJWVBIiM1IiZsXiQiIzctSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

coeff(coeff(convert(Cser(n),polynom),x,2),y,-2);

LCoqJiImbF4kIiIiKUkidEc2IiIjNkYlRiUqJiIlZUFGJSlGJyIiKkYlRiUqJiIkTCJGJSlGJyIiKEYlRiUqJiIiJ0YlKUYnIiImRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-3): the series B12 (in Q(T))</Text-field>

collect(map(Z->convert(series(Z,y,3),polynom),convert(series(subs(Ker=K,eqB),x,3),polynom)),[x,y],distributed,factor);

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

coeff(coeff(%,x,1),y,1);

LCwqJkkidEc2IiIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlRiZGJiIiIzYjSSJCR0YlNiQiIiFGM0YmRiYqJkYkRiYtLSZGKjYlRiZGL0YvRjBGMkYmRiYqJkYvRiYtLSZGKjYkRiZGL0YwRjJGJiEiIiooRi9GJkYkRiYtLSZGKjYjRiZGMEYyRiZGJiooRi9GJkYkRiYtLSZGKjYjRi9GMEYyRiZGJg==

expand(isolate(%,(D[1,2, 2](B))(0, 0)));

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlIiIiIiIjRi02I0kiQkdGKjYkIiIhRjEsKi0tJkYmNiVGLEYsRi1GLkYwISIiKihGLUYsSSJ0R0YqRjctLSZGJjYkRixGLUYuRjBGLEYsKiZGLUYsLS0mRiY2I0YsRi5GMEYsRjcqJkYtRiwtLSZGJjYjRi1GLkYwRixGNw==

B12ftZm:=factor(-B21ftZm+B11Zm-B10ftZm-B01ftZm);

KiopLCZJI1ptRzYiIiIiRichIiIiIiRGJywqKiQpRiUiIidGJ0YnKiYiI1ZGJylGJSIiJUYnRigqJiIkSiJGJylGJSIiI0YnRiciJG4iRidGJyksJkYlRidGJ0YnRilGJyksJiokRjRGJ0YnRilGJ0YtRig=

C1m3ftsol:=B12ftZm:

n:=12: normal(series(subs(Zm=Zmser(n),C1m3ftsol),t,n+1));

Ky1JInRHNiIiIiUiIiciIygpIiIpIiVnOSIjNSImaUQjIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,1),y,-3);

LCoqJiImaUQjIiIiKUkidEc2IiIjNkYlRiUqJiIlZzlGJSlGJyIiKkYlRiUqJiIjKClGJSlGJyIiKEYlRiUqJiIiJUYlKUYnIiImRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0): the series P00 (in Q(T))</Text-field>

limit(limit(Pexpr,x=0),y=0);

LCYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIjYjSSJMR0YqNiQiIiFGMEYsLS0mRiY2IyIiIzYjSSJCR0YqRi9GLA==

already found when studying walks starting from (0,0) and ending at (-1,0)

P00ftZm:=factor(L10ftZm+B01ftZm);

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

C00ftsol:=P00ftZm;

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

n:=12: normal(series(subs(Zm=Zmser(n),C00ftsol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIigiIiUiI2wiIiciJCNwIiIpIiUiKnoiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

coeff(coeff(convert(Cser(n),polynom),x,0),y,0);

LC4qJiImLXUqIiIiKUkidEc2IiIjNkYlRiUqJiIlIip6RiUpRiciIipGJUYlKiYiJCNwRiUpRiciIihGJUYlKiYiI2xGJSlGJyIiJkYlRiUqJkYxRiUpRiciIiRGJUYlRidGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,0): the series P10</Text-field>

series(coeff(series(Pexpr,x,3),x,1),y,2);

KydJInlHNiIsJiomIyIiIiIiI0YoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRGKEYoNiNJIkxHRiQ2JCIiIUY1RihGKC0tJkYtNiRGKEYpNiNJIkJHRiRGNEYoRjUtSSJPR0YvNiNGKEYo

P10Zm:=factor(L20Zm+B11Zm);

LCQqLiIjSyIiIiwmSSNabUc2IkYlRiUhIiJGJSlGJyIiJEYlLDQqJClGJyIiKUYlRiUqJiIjNUYlKUYnIiIoRiVGKSomRi9GJSlGJyIiJ0YlRiUqJiIjPUYlKUYnIiImRiVGKSomIiMjKUYlKUYnIiIlRiVGJSomIiM5RiVGKkYlRikqJiIkNyJGJSlGJyIiI0YlRiUqJiIjQUYlRidGJUYpIiM2RilGJSksJiokRkNGJUYlRitGJUY2RiksJkYnRiVGJUYlRilGJQ==

C10sol:=P10Zm:

n:=12: normal(series(subs(Zm=Zmser(n),C10sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIzgiIiUiJGYiIiInIiUkKT4iIikiJiVSRCIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

coeff(coeff(convert(Cser(n),polynom),x,1),y,0);

LC4qJiInOUlMIiIiKUkidEc2IiIjN0YlRiUqJiImJVJERiUpRiciIzVGJUYlKiYiJSQpPkYlKUYnIiIpRiVGJSomIiRmIkYlKUYnIiInRiVGJSomIiM4RiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,1): the series P01</Text-field>

series(Pexpr,x,2);;

KylJInhHNiIsKComLUkiQkdGJDYkIiIhRioiIiJJInlHRiQhIiJGLS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRis2I0kiTEdGJDYkRipGLEYrKiYtRihGOEYrRixGLUYrRiosKComLS1GMDYjRihGKUYrRixGLUYtKiYjRisiIiNGKy0tJkYxNiRGK0YrRjZGOEYrRisqJi1GPkY4RitGLEYtRitGKy1JIk9HRjNGNUZC

series(coeff(series(Pexpr,x,2),x,0),y,3);

KylJInlHNiIsJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiNiNJIkxHRiQ2JCIiIUYyRi4tLSZGKTYjIiIjNiNJIkJHRiRGMUYuRjIsJi0tJkYpNiRGLkY3Ri9GMUYuKiYjRi5GN0YuLS0mRik2JEY3RjdGOEYxRi5GLkYuLUkiT0dGK0YtRjc=

P01Zm:=factor(L11Zm+B02Zm);

LCQqLiIjayIiIiwmSSNabUc2IkYlRiUhIiJGJSlGJyIiJEYlLDAqJiIiKkYlKUYnIiInRiVGJSomIiM5RiUpRiciIiZGJUYpKiZGK0YlKUYnIiIlRiVGJSomIiNXRiVGKkYlRikqJiIjREYlKUYnIiIjRiVGKSomIiNRRiVGJ0YlRikiIz5GKUYlKSwmKiRGPEYlRiVGK0YlRjBGKSwmRidGJUYlRiVGKUYp

C01sol:=P01Zm:

n:=12: normal(series(subs(Zm=Zmser(n),C01sol),t,n+1));

Ky9JInRHNiIiIiNGJSIjQCIiJSIkSyMiIiciJUFGIiIpIiZZTSQiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

coeff(coeff(convert(Cser(n),polynom),x,0),y,1);

LC4qJiInImZEJSIiIilJInRHNiIiIzdGJUYlKiYiJllNJEYlKUYnIiM1RiVGJSomIiVBRkYlKUYnIiIpRiVGJSomIiRLI0YlKUYnIiInRiVGJSomIiNARiUpRiciIiVGJUYlKiYiIiNGJSlGJ0Y7RiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,1): the series P11 (in Q(T))</Text-field>

series(coeff(series(Pexpr,x,3),x,1),y,3);

KylJInlHNiIsJiomIyIiIiIiI0YoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRGKEYoNiNJIkxHRiQ2JCIiIUY1RihGKC0tJkYtNiRGKEYpNiNJIkJHRiRGNEYoRjUsJiomRidGKC0tJkYtNiVGKEYoRilGMkY0RihGKComRidGKC0tJkYtNiVGKEYpRilGOkY0RihGKEYoLUkiT0dGLzYjRihGKQ==

P11ftZm:=factor(L21ftZm+B12ftZm);

KiopLCZJI1ptRzYiIiIiRichIiIiIiNGJyksJkYlRidGJ0YnRilGJywsKiQpRiUiIilGJ0YnKiYiI1dGJylGJSIiJ0YnRigqJiIkNSJGJylGJSIiJUYnRicqJiIkPyVGJylGJUYpRidGJyIkIkdGJ0YnKSwmKiRGOkYnRiciIiRGJ0YzRig=

C11ftsol:=P11ftZm:

n:=12: normal(series(subs(Zm=Zmser(n),C11ftsol),t,n+1));

Ky9JInRHNiIiIiQiIiUiI1ciIiciJCJlIiIpIiVHdyIjNSInPjg1IiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

coeff(coeff(convert(Cser(n),polynom),x,1),y,1);

LCwqJiInPjg1IiIiKUkidEc2IiIjNkYlRiUqJiIlR3dGJSlGJyIiKkYlRiUqJiIkImVGJSlGJyIiKEYlRiUqJiIjV0YlKUYnIiImRiVGJSomIiIkRiUpRidGN0YlRiU=

<Text-field style="Heading 2" layout="Heading 2">Looking for nice numbers</Text-field>

restart:

algTm:=256*Tm^3*t^2-Tm^4-8*Tm^3-18*Tm^2+27;

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

algZm:=subs(Tm=Zm^2,algTm);

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

with(gfun):Tmser:=proc(n) op(1,algeqtoseries(algTm,t,Tm,n,true)): end:

Tmser(5);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

Zmser:=proc(n) series(sqrt(Tmser(n)),t,n+1): end:

with(LREtools):

<Text-field style="Heading 3" layout="Heading 3">What about walks ending at (-1,0) ? Hypergeometric</Text-field>

Cm10sol :=64*Zm^3/(Zm^2+3)^3;

LCQqKCIjayIiIilJI1ptRzYiIiIkRiUpLCYqJClGJyIiI0YlRiVGKUYlRikhIiJGJQ==

n:=4:series(subs(Zm=Zmser(n),Cm10sol),t,n+1);

KylJInRHNiIiIiIiIiEiIiQiIiMtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIiU=

algCm10:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm10-Cm10sol),algZm,Zm))));

LDgqKCI5JXlJIWZmXT1KPFwhNCIiIiIpSSVDbTEwRzYiIiIpRiUpSSJzR0YoIiInRiVGJSooIjk7TyVccUBebCc+bFE+RiUpRidGLEYlKUYrIiImRiVGJSooIjh3KmVsNVg0enVsNjdGJUYvRiUpRisiIiVGJUYlKigiOUd0cSFRSiQqXHpJKVs2RiUpRidGNUYlRjRGJUYlKigiOC86JWZFbiEqcCg+R2cjRiVGOEYlKUYrIiIkRiUhIiIqKCI2KSllP2pVSEg/d1slRiVGOEYlKUYrIiIjRiVGJSooIjgzJ0chcGw8dlImSHBBRiUpRidGQUYlRjtGJUYlKigiNjNtJDM/JHozaCc9YEYlRkRGJUZARiVGPSooIjUpR19dZCp6SjtDTEYlRkRGJUYrRiVGPSomIjRbPyVlZm0+RlNiRiVGREYlRiVGSkY9

algeqtodiffeq(algCm10,Cm10(s));

PCQsLComLCgqJiImISlvIyIiIilJInNHNiIiIiNGKEYoKiYiJTNNRihGKkYoISIiIiQ/IkYoRigtSSVDbTEwR0YrNiNGKkYoRigqJiwqKiYiJ1NBX0YoKUYqIiIkRihGKComIiZLZyhGKEYpRihGLyomIiVvTEYoRipGKEYoIiNTRi9GKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGMUYqRihGKComLCoqJiInZzgiKkYoKUYqIiIlRihGKComIidXcDlGKEY4RihGLyomIiVzd0YoRilGKEYoKiYiJEciRihGKkYoRi9GKC1GQDYkRj9GKkYoRigqJiwqKiYiJ1MnbyRGKClGKiIiJkYoRigqJiImN1gnRihGR0YoRi8qJiIlV1BGKEY4RihGKComIiNzRihGKUYoRi9GKC1GQDYkRk9GKkYoRigqJiwqKiYiJmtvJEYoKUYqIiInRihGKComIiU3cEYoRlVGKEYvKiYiJEslRihGR0YoRigqJiIiKkYoRjhGKEYvRigtRkA2JEZnbkYqRihGKC8tRjI2IyIiIS1JJ1Jvb3RPZkc2JEZBSShfc3lzbGliR0YrNiMsJiokKUkjX1pHRl1wRixGKEYoRihGLw==

de:={op(1,%),Cm10(0)=1};

rec:=diffeqtorec(de,Cm10(s),co(k));

PCYsKiomLCwqJiIma28kIiIiKUkia0c2IiIiJUYoRigqJiInY3U5RigpRioiIiRGKEYoKiYiJ1c0QEYoKUYqIiIjRihGKComIid3cDdGKEYqRihGKCImISlvI0YoRigtSSNjb0dGKzYjRipGKEYoKiYsLComIiU3cEYoRilGKCEiIiomIiYpb11GKEYvRihGPyomIidLKzlGKEYzRihGPyomIicpR3MiRihGKkYoRj8iJlMlekY/RigtRjk2IywmRipGKEYoRihGKEYoKiYsLComIiRLJUYoRilGKEYoKiYiJTNZRihGL0YoRigqJiImcyU9RihGM0YoRigqJiImM0kkRihGKkYoRigiJitBI0YoRigtRjk2IywmRipGKEY0RihGKEYoKiYsLComIiIqRihGKUYoRj8qJiIkRSJGKEYvRihGPyomIiRmJ0YoRjNGKEY/KiYiJUU6RihGKkYoRj8iJT84Rj9GKC1GOTYjLCZGKkYoRjBGKEYoRigvLUY5NiMiIiFGKC8tRjk2I0YoRjAvLUY5NiNGNCIjQA==

rec1:=map(factor,op(1,%));

LCoqLiIkYyMiIiIsJiomIiInRiVJImtHNiJGJUYlIiImRiVGJSwmKiYiIiNGJUYpRiVGJSIiJEYlRiUsJkYtRiVGJUYlRiUsJkYnRiUiIihGJUYlLUkjY29HRio2I0YpRiVGJSoqIiM7RiVGLEYlLCoqJiIkOyNGJSlGKUYvRiVGJSomIiVnN0YlKUYpRi5GJUYlKiYiJSdbI0YlRilGJUYlIiViO0YlRiUtRjQ2IywmRilGJUYlRiVGJSEiIioqIiIpRiUsJkYpRiVGL0YlRiUsKiomIiNhRiVGO0YlRiUqJiIkOSVGJUY+RiVGJSomIiVuNUYlRilGJUYlIiREKkYlRiUtRjQ2IywmRilGJUYuRiVGJUYlKiwsJiomRi9GJUYpRiVGJSIjNUYlRiUsJkYpRiUiIiVGJUYlRkhGJSwmRlZGJSIjNkYlRiUtRjQ2I0ZIRiVGRQ==

HG solution

with(LREtools):map(factor,hypergeomsols(rec1,co(k),{co(0) = 1, co(1) = 3, co(2) = 21}));

LCYqNiMiIiUiIioiIiItSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiIyIiJCEiIilGMSNGJ0YwRicpSSNQaUdGK0Y0RjItRik2IyMiIiYiIidGJykiIztJImtHRi1GJy1GKTYjLCZGPkYnRjRGJ0YnLUYpNiMsJkY+RicjIiIoRjtGJ0YnLUYpNiMsJkY+RicjRiVGMUYnRjItRik2IywmRj5GJ0YwRidGMkYnKjQjRjBGJkYnRjdGMkY1RjJGKEYnRjxGJ0Y/RictRik2IywmRj5GJ0Y5RidGJy1GKTYjLCZGPkYnI0Y6RjFGJ0YyRktGMkYn

In fact there seems to be a shorter recurrence relation

n:=70: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm10sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

PCUsKComLC4qJiIlOyMqIiIiKUkia0c2IiIiJkYoRigqJiImK3cmRigpRioiIiVGKEYoKiYiJyFvTiJGKClGKiIiJEYoRigqJiInKy86RigpRioiIiNGKEYoKiYiJlcieUYoRipGKEYoIiY/XiJGKEYoLUkjY29HRis2I0YqRihGKComLC4qJiIlXzZGKEYpRighIiIqJiImITM1RihGL0YoRkMqJiImISlbJEYoRjNGKEZDKiYiJlMnZkYoRjdGKEZDKiYiJm8uJkYoRipGKEZDIiYrbyJGQ0YoLUY9NiMsJkYqRihGKEYoRihGKComLC4qJiIjT0YoRilGKEYoKiYiJDAlRihGL0YoRigqJiIlIXoiRihGM0YoRigqJiIldlFGKEY3RihGKComIiUlNCVGKEYqRihGKCIlIW8iRihGKC1GPTYjLCZGKkYoRjhGKEYoRigvLUY9NiMiIiFGKC8tRj02I0YoRjQ=

rec1:=map(factor,op(1,rec));

LCgqMCIjOyIiIiwmKiYiIidGJUkia0c2IkYlRiUiIiZGJUYlLCYqJiIiI0YlRilGJUYlIiIkRiVGJSwmRi1GJUYlRiVGJSwmRidGJSIiKEYlRiUsJiomIiIlRiVGKUYlRiUiIipGJUYlLUkjY29HRio2I0YpRiVGJSouIiIpRiVGLEYlLCZGNEYlRjJGJUYlLCZGKUYlRi5GJUYlLCgqJiIjPUYlKUYpRi5GJUYlKiYiI2pGJUYpRiVGJSIjXUYlRiUtRjg2IywmRilGJUYlRiVGJSEiIiouLCZGNEYlRitGJUYlLCYqJkYvRiVGKUYlRiVGMkYlRiUsJkYpRiVGL0YlRiVGPUYlLCZGTEYlRjtGJUYlLUY4NiNGPUYlRiU=

(hypergeomsols(rec1,co(k),{co(0) = 1, co(1) = 3}));

fact:=proc(a,n) if n=0 then 1 else a*fact(a+1,n-1): fi: end:

seq((4/9)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+7/6)*GAMMA(k+1/2)/(GAMMA(2/3)*sqrt(Pi)*GAMMA(k+2)*GAMMA(k+4/3))/ (fact(1/2,k)*fact(7/6,k)*16^k/fact(2,k)/fact(4/3,k)),k=1..10);

NiwjIiIjIiIkRiNGI0YjRiNGI0YjRiNGI0Yj

seq((2/9)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)/(GAMMA(5/6)*sqrt(Pi)*GAMMA(k+2)*GAMMA(k+5/3))/ (fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k)*16^k), k=1..10);

NiwjIiIiIiIkRiNGI0YjRiNGI0YjRiNGI0Yj

paper version

seq(16^k/3*(2*fact(1/2,k)*fact(7/6,k)/fact(2,k)/fact(4/3,k)+fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k)),k=0..10);

Ni0iIiIiIiQiI0AiJCQ+IiVNPyImJEdCIicpbyJHIig+ZmEkIiljISkpZiUiKkknZi9oIitQcHZiIyk=

coeff(coeff(convert(Cser(20),polynom),x,-1),y,0);

LDgqJiIrUHB2YiMpIiIiKUkidEc2IiIjP0YlRiUqJiIqSSdmL2hGJSlGJyIjPUYlRiUqJiIpYyEpKWYlRiUpRiciIztGJUYlKiYiKD5mYSRGJSlGJyIjOUYlRiUqJiInKW8iR0YlKUYnIiM3RiVGJSomIiYkR0JGJSlGJyIjNUYlRiUqJiIlTT9GJSlGJyIiKUYlRiUqJiIkJD5GJSlGJyIiJ0YlRiUqJiIjQEYlKUYnIiIlRiVGJSomIiIkRiUpRiciIiNGJUYlRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,1) : no</Text-field>

Cm11ftsol:=-16*Zm^2*(Zm-1)*(Zm-3)/(Zm^2+3)^3;

LCQqLCIjOyIiIilJI1ptRzYiIiIjRiUsJkYnRiVGJSEiIkYlLCZGJ0YlIiIkRitGJSksJiokRiZGJUYlRi1GJUYtRitGKw==

n:=10: series(subs(Zm=Zmser(n),Cm11ftsol),t,n+1);

Ky1JInRHNiIiIiIiIiMiIikiIiUiI3giIiciJEkpRictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

algCm11ft:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm11ft-Cm11ftsol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm11ft,Cm11ft(s)): nops(%);

IiIk

op(2,de0);op(3,de0);

Ly1JJ0NtMTFmdEc2IjYjIiIhLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiokKUkjX1pHRioiIiMiIiJGMyomRjJGM0YxRjNGMw==

Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ0NtMTFmdEdGKTYjIiIhLCYqJiIiJCIiIi1JJ1Jvb3RPZkdGJjYjLCYqJClJI19aR0YmIiIjRjFGMSomRjlGMUY4RjFGMUYxRjFGMUYx

de:={op(1,de0),Cm11ft(0)=0, D(Cm11ft)(0)=1}:

rec:=diffeqtorec(de,Cm11ft(s),co(k)):

rec1:=map(factor,op(1,%));

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

{seq(op(i,rec),i=2..nops(rec))};

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIikvLUYlNiMiIiQiI3gvLUYlNiMiIiUiJEkpLy1GJTYjIiImIiVEJyovLUYlNiMiIiciJzV1Ni8tRiU2IyIiKCIoSmlbIi8tRiU2I0YxIilDPE4+Ly1GJTYjIiIqIiphd2lkIw==

Some HG solutions, but not ours

hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1, co(2) = 8, co(3) = 77, co(4) = 830, co(5) = 9625, co(6) = 117410, co(7) = 1486231, co(8) = 19351724, co(9) = 257627654});

hypergeomsols(rec1,co(k),{});

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGTkYnRj9GJ0Y5RidGLEY5RkJGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRictRi02IywmRkZGJyNGNEYpRidGOUZTRjlGOQ==

In fact there seems to be a shorter recurrence relation

n:=180: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm11ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1,co(2) = 8, co(3) = 77}));

hypergeomsols(rec1,co(k),{});

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmKiZGK0YnLUkjY29HRjE2IyIiIUYnRictRj02I0YnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnIyIiKEY1RidGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRidGK0YnRjktRi02IywmRkZGJyNGJUYpRidGOUY5KjYjRitGJkYnRjZGJywmKiZGS0YnRjxGJ0YnKiZGK0YnRkBGJ0Y5RidGLEY5RkJGOUZERidGTEYnLUYtNiMsJkZGRidGM0YnRidGT0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGJw==

hypergeomsols(rec1,co(k),{co(2) = 8, co(3) = 77});

LCYqNiMiJE0kIiVAOiIiIi1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiIjIiIkISIiKUYxI0YnRjBGJylJI1BpR0YrRjRGMi1GKTYjIyIiJiIiJ0YnKSIjO0kia0dGLUYnLUYpNiMsJkY+RicjIiIoRjtGJ0YnLUYpNiMsJkY+RidGNEYnRictRik2IywmRj5GJ0YwRidGMi1GKTYjLCZGPkYnIyIiJUYxRidGMkYnKjQjRk4iJDwiRidGN0YyRjVGMkYoRidGPEYnRkRGJy1GKTYjLCZGPkYnRjlGJ0YnRkdGMi1GKTYjLCZGPkYnI0Y6RjFGJ0YyRjI=

seq(%,k=2..10);

NisiIikiI3gjIicnSFQiIiRwIiMiKHBSSSQiJFEkIyIpVXRJP0YnIyIpPkcjKVIiI0UjIipBaiczRSIjOCMiK1B2LyNcJEYyIyIsUTRzaXYlRjI=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0) : no</Text-field>

Cm20ftsol:=(16*(Zm-1))*(Zm+1)/(Zm^2+3)^3;

LCQqKiIjOyIiIiwmSSNabUc2IkYlRiUhIiJGJSwmRidGJUYlRiVGJSksJiokKUYnIiIjRiVGJSIiJEYlRjBGKUYl

n:=12: series(subs(Zm=Zmser(n),Cm20ftsol),t,n+1);

Ky9JInRHNiIiIiIiIiMiIiciIiUiI15GJyIkNyYiIikiJW5jIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

algCm20ft:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm20ft-Cm20ftsol),algZm,Zm)))):

algeqtodiffeq(algCm20ft,Cm20ft(s));

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

de:=%:

rec:=diffeqtorec(de,Cm20ft(s),co(k)):

rec1:=map(factor,op(1,%));

LCwqLCImT2InIiIiSSJrRzYiRiUsJiomIiIkRiVGJkYlRiUiIiNGJUYlLCZGKUYlRiVGJUYlLUkjY29HRic2I0YmRiVGJSooIiVDNUYlLCoqJiIjIilGJSlGJkYqRiVGJSomIiQoSEYlKUYmRitGJUYlKiYiJFAkRiVGJkYlRiUiJDgiRiVGJS1GLjYjLCZGJkYlRiVGJUYlISIiKioiIycqRiUsJiomRitGJUYmRiVGJSIiJkYlRiUsKComIiNJRiVGOEYlRiUqJiIkXSJGJUYmRiVGJSIkIj1GJUYlLUYuNiMsJkYmRiVGK0YlRiVGJSooIiIlRiUsKkYzRiUqJiIkPSpGJUY4RiVGJSomIiVVTUYlRiZGJUYlIiVzVUYlRiUtRi42IywmRiZGJUYqRiVGJUY/KiosJkYmRiVGREYlRiUsJkYpRiUiIzlGJUYlLCZGKUYlIiM4RiVGJS1GLjYjLCZGJkYlRk9GJUYlRiU=

{seq(op(i,rec),i=2..nops(rec))};

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIicvLUYlNiMiIiQiI14=

No HG solution

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

IiIh

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm20ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

PCUsKComLCwqJiIlL0IiIiIpSSJrRzYiIiImRihGKComIiY/OiJGKClGKiIiJUYoRigqJiImVyo9RigpRioiIiRGKEYoKiYiJms3IkYoKUYqIiIjRihGKComIiVbP0YoRipGKEYoRigtSSNjb0dGKzYjRipGKEYoKiYsLiomIiQpR0YoRilGKCEiIiomIiVnQEYoRi9GKEZCKiYiJUdoRihGM0YoRkIqJiIlTyIpRihGN0YoRkIqJiIlP1xGKEYqRihGQiIlMzVGQkYoLUY8NiMsJkYqRihGKEYoRihGKComLC4qJiIiKkYoRilGKEYoKiYiIyEqRihGL0YoRigqJiIkVyRGKEYzRihGKComIiRBJ0YoRjdGKEYoKiYiJEYmRihGKkYoRigiJG8iRihGKC1GPDYjLCZGKkYoRjhGKEYoRigvLUY8NiMiIiFGXG8vLUY8NiNGKEYo

rec1:=map(factor,op(1,rec));

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(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1}));

hypergeomsols(rec1,co(k),{});

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,0) : no</Text-field>

Cm30sol:=-512*Zm^3*(Zm-1)*(Zm^3-3*Zm^2-Zm-1)/(Zm^2+3)^6;

LCQqLCIkNyYiIiIpSSNabUc2IiIiJEYlLCZGJ0YlRiUhIiJGJSwqKiRGJkYlRiUqJkYpRiUpRiciIiNGJUYrRidGK0YlRitGJSksJiokRi9GJUYlRilGJSIiJ0YrRis=

n:=12: series(subs(Zm=Zmser(n),Cm30sol),t,n+1);

Ky9JInRHNiIiIiIiIiMiIzUiIiUiJDEiIiInIiUuNyIiKSImKVE5RictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

algCm30:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm30-Cm30sol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm30,Cm30(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUNtMzBHNiI2IyIiIS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqJClJI19aR0YqIiIjIiIiRjNGMSEiIg==

de:={op(1,de0),Cm30(0)=0}: :

rec:=diffeqtorec(de,Cm30(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PC8vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIzUvLUYlNiMiIiQiJDEiLy1GJTYjIiIlIiUuNy8tRiU2IyIiJiImKVE5Ly1GJTYjIiInIidYIXoiLy1GJTYjIiIoIihNdEgjLy1GJTYjIiIpIilsJyo+SS8tRiU2IyIiKiIqQlQkW1MvLUYlNiNGMSIrQiNIXV4mLy1GJTYjIiM2IixFU0ldaCgvLUYlNiMiIzciLmA5I3BiajU=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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In fact there seems to be a shorter recurrence relation

n:=180: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm30sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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rec1:=map(factor,op(1,rec));

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(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1,co(2) = 10, co(3) = 106}));

hypergeomsols(rec1,co(k),{});

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(hypergeomsols(rec1,co(k),{co(2) = 10, co(3) = 106}));

LCYqOCMiIyMpIiQqPSIiIi1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiIjIiIkISIiKUYxI0YnRjBGJylJI1BpR0YrRjRGMi1GKTYjIyIiJiIiJ0YnKSIjO0kia0dGLUYnLUYpNiMsJkY+RicjIiIoRjtGJ0YnLUYpNiMsJkY+RidGNEYnRictRik2IywmRj5GJ0YxRidGMi1GKTYjLCZGPkYnI0ZDRjFGJ0YyLCgqJClGPkYwRidGJyomI0Y6RjBGJ0Y+RidGJyMiIzwiIzdGJ0YnRicqNiMiJCZIIiVyNUYnRjdGMkYoRidGNUYyRjxGJy1GKTYjLCZGPkYnIyIjNkY7RidGJ0ZERidGR0YyLUYpNiMsJkY+RicjIiIpRjFGJ0YyLCZGPkYnI0Y6RjFGJ0YnRjI=

seq(%,k=2..10);

NisiIzUiJDEiIyInJnlAKSIkcycjIik6Uj45IiRfKiMiKmwieSg+KCIlM1EjIisoZW5nLygiJWNHIyIsYj0+N0YnIiUvPiMiLFp1Vk9EIiIjRyMiLipHZyVbYDIjIiRPJA==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,2) : no</Text-field>

Cm12sol := 64*Zm^3*(Zm-1)*(9*Zm^6-78*Zm^5+123*Zm^4+36*Zm^3+7*Zm^2-22*Zm-11)/((Zm^2+3)^6*(Zm+1));

n:=12: series(subs(Zm=Zmser(n),Cm12sol),t,n+1);

Ky9JInRHNiIiIiIiIiMiIzoiIiUiJCM+IiInIiVSQyIiKSImdzkkIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

algCm12:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm12-Cm12sol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm12,Cm12(s)): nops(%);

IiIp

op(2,de0);

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de:=de0: :

rec:=diffeqtorec(de,Cm12(s),co(k)):

rec1:=map(factor,rec):

nops(rec1);

IiM5

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),{co(0)=0,co(1)=1, co(2)=15}));

hypergeomsols(rec1,co(k),{});

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In fact there seems to be a shorter recurrence relation

n:=180: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm12sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1, co(2) = 15, co(3) = 192}));

hypergeomsols(rec1,co(k),{});

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

(hypergeomsols(rec1,co(k),{co(2) = 15, co(3) = 192}));

LCYqOCMiJiNSSSImRCM9IiIiKUkjUGlHJSpwcm90ZWN0ZWRHI0YnIiIjISIiKSIiJEYrRictSSZHQU1NQUc2JEYqSShfc3lzbGliRzYiNiMjRixGL0YtLUYxNiMjIiImIiInRicpIiM7SSJrR0Y0RictRjE2IywmRj5GJ0YrRidGJy1GMTYjLCZGPkYnIyIiKEY7RidGJy1GMTYjLCZGPkYnI0ZGRi9GJ0YtLUYxNiMsJkY+RidGL0YnRi0sKCokKUY+RixGJ0YnKiYjIiNdIiNIRidGPkYnRicjIiNjIiMoKUYnRidGJyo2IyIoXyo+VSIoRCUpeSNGJ0Y3Ri1GKEYtRjBGJ0Y8RictRjE2IywmRj5GJ0Y5RidGJ0Y/RictRjE2IywmRj5GJyMiIilGL0YnRi1GS0YtLCgqJiMiJCIpKiIkUyVGJ0ZQRidGJyomIyIkMiMiIykpRidGPkYnRicjRi8iIiVGJ0YnRi0=

seq(%,k=2..10);

NisiIzoiJCM+IyIpbSZbZiIiJXZqIyIpJ3BvMCgiJURAIyIqXzF0XypGKiMiLClSJjNoIlJGJyMiLSUpeStDTWFGJyMiLWo5LFJwKikiJF0oIyIvKCl5SzlhMD4iJUQ2

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-1) : HyperGeometric</Text-field>

C0m1sol := -32*Zm^3*(Zm-1)*(Zm^2-2*Zm-1)/((Zm+1)*(Zm^2+3)^3);

LCQqLiIjSyIiIilJI1ptRzYiIiIkRiUsJkYnRiVGJSEiIkYlLCgqJClGJyIiI0YlRiUqJkYvRiVGJ0YlRitGJUYrRiUsJkYnRiVGJUYlRispLCZGLUYlRilGJUYpRitGKw==

n:=12: series(subs(Zm=Zmser(n),C0m1sol),t,n+1);

Ky9JInRHNiIiIiIiIiMiIzUiIiUiJDMiIiInIiVfNyIiKSImel8iRictSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

algC0m1:=factor(subs(t=sqrt(s),factor(resultant(numer(C0m1-C0m1sol),algZm,Zm)))):

de0:=algeqtodiffeq(algC0m1,C0m1(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUMwbTFHNiI2IyIiIUYn

de:=de0: :

rec:=diffeqtorec(de,C0m1(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PCUvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIzU=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

LCYqNiMiIiUiIioiIiItSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiIyIiJCEiIilGMSNGJ0YwRicpSSNQaUdGK0Y0RjItRik2IyMiIiYiIidGJykiIztJImtHRi1GJy1GKTYjLCZGPkYnRjRGJ0YnLUYpNiMsJkY+RicjIiIoRjtGJ0YnLUYpNiMsJkY+RicjRiVGMUYnRjItRik2IywmRj5GJ0YwRidGMkYnKjRGJEYnRjdGMkYoRidGNUYyRjxGJ0Y/RictRik2IywmRj5GJ0Y5RidGJy1GKTYjLCZGPkYnI0Y6RjFGJ0YyRktGMkYy

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C0m1sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

rec1:=map(factor,op(1,rec));

map(factor,hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1}));

LCYqNiMiIiUiIioiIiItSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiIyIiJCEiIilGMSNGJ0YwRicpSSNQaUdGK0Y0RjItRik2IyMiIiYiIidGJykiIztJImtHRi1GJy1GKTYjLCZGPkYnRjRGJ0YnLUYpNiMsJkY+RicjIiIoRjtGJ0YnLUYpNiMsJkY+RicjRiVGMUYnRjItRik2IywmRj5GJ0YwRidGMkYnKjRGJEYnRjdGMkYoRidGNUYyRjxGJ0Y/RictRik2IywmRj5GJ0Y5RidGJy1GKTYjLCZGPkYnI0Y6RjFGJ0YyRktGMkYy

seq((4/9)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+7/6)*GAMMA(k+1/2)/(GAMMA(2/3)*sqrt(Pi)*GAMMA(k+2)*GAMMA(k+4/3))/ (fact(1/2,k)*fact(7/6,k)*16^k/fact(2,k)/fact(4/3,k)),k=1..10);

NiwjIiIjIiIkRiNGI0YjRiNGI0YjRiNGI0Yj

seq((4/9)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)/(GAMMA(5/6)*sqrt(Pi)*GAMMA(k+2)*GAMMA(k+5/3))/ (fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k)*16^k), k=1..10);

NiwjIiIjIiIkRiNGI0YjRiNGI0YjRiNGI0Yj

paper version

seq(2*16^k/3*(fact(1/2,k)*fact(7/6,k)/fact(2,k)/fact(4/3,k)-fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k)),k=0..10);

Ni0iIiEiIiIiIzUiJDMiIiVfNyImel8iIidXTz4iKGRkXyMiKSd6JHBMIiphO25kJSIrcWtXNGo=

coeff(coeff(convert(Cser(20),polynom),x,0),y,-1);

LDYqJiIrcWtXNGoiIiIpSSJ0RzYiIiM/RiVGJSomIiphO25kJUYlKUYnIiM9RiVGJSomIikneiRwTEYlKUYnIiM7RiVGJSomIihkZF8jRiUpRiciIzlGJUYlKiYiJ1dPPkYlKUYnIiM3RiVGJSomIiZ6XyJGJSlGJyIjNUYlRiUqJiIlXzdGJSlGJyIiKUYlRiUqJiIkMyJGJSlGJyIiJ0YlRiUqJkY9RiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-2) : no</Text-field>

C0m2ftsol;

LCQqKiwmKiQpSSNabUc2IiIiIyIiIkYqIiImISIiRiopLCZGJ0YqRipGLEYpRiopLCZGJ0YqRipGKkYpRiopLCZGJUYqIiIkRipGM0YsRiw=

C0m2ftsol := -(Zm^2-5)*(Zm-1)^2*(Zm+1)^2/(Zm^2+3)^3;

LCQqKiwmKiQpSSNabUc2IiIiIyIiIkYqIiImISIiRiopLCZGJ0YqRipGLEYpRiopLCZGJ0YqRipGKkYpRiopLCZGJUYqIiIkRipGM0YsRiw=

n:=12: series(subs(Zm=Zmser(n),C0m2ftsol),t,n+1);

Ky9JInRHNiIiIiIiIiUiIzkiIiciJCE9IiIpIiVDQiIjNSImRTAkIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlRic=

algC0m2ft:=factor(subs(t=sqrt(s),factor(resultant(numer(C0m2ft-C0m2ftsol),algZm,Zm)))):

de0:=algeqtodiffeq(algC0m2ft,C0m2ft(s)); nops(%);

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

IiIn

de:=de0: :

rec:=diffeqtorec(de,C0m2ft(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGLC8tRiU2IyIiJCIjOQ==

No HG solution

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

IiIh

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C0m2ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

PCYsKComLCwqJiIlL0IiIiIpSSJrRzYiIiImRihGKComIiY/OiJGKClGKiIiJUYoRigqJiImU20iRigpRioiIiRGKEYoKiYiJWcqKUYoKUYqIiIjRihGKComIiVPOkYoRipGKEYoRigtSSNjb0dGKzYjRipGKEYoKiYsLiomIiQpR0YoRilGKCEiIiomIiVnQEYoRi9GKEZCKiYiJVNlRihGM0YoRkIqJiIlU29GKEY3RihGQiomIiU3SkYoRipGKEZCIiRTI0ZCRigtRjw2IywmRipGKEYoRihGKEYoKiYsLComIiIqRihGKUYoRigqJiIjISpGKEYvRihGKComIiROJEYoRjNGKEYoKiYiJF0mRihGN0YoRigqJiIkTyRGKEYqRihGKEYoLUY8NiMsJkYqRihGOEYoRihGKC8tRjw2IyIiIUZbby8tRjw2I0YoRltvLy1GPDYjRjhGKA==

rec1:=map(factor,op(1,rec));

LCgqMCIkYyMiIiJJImtHNiJGJSwmKiYiIiRGJUYmRiVGJSIiI0YlRiUsJkYmRiVGKkYlRiUsJkYpRiVGJUYlRiUsJkYmRiVGJUYlRiUtSSNjb0dGJzYjRiZGJUYlKi4iIilGJSwmKiZGK0YlRiZGJUYlRipGJUYlLCZGJkYlRitGJUYlRi5GJSwoKiYiIz1GJSlGJkYrRiVGJSomIiNhRiVGJkYlRiUiIiZGJUYlLUYwNiNGLkYlISIiKi5GJkYlLCZGKUYlIiIoRiVGJUYsRiVGNkYlLCZGKUYlRjNGJUYlLUYwNiNGNkYlRiU=

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 1}));

hypergeomsols(rec1,co(k),{});

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-3): no</Text-field>

C0m3sol;

LCQqLiIkNyYiIiIpSSNabUc2IiIiJkYlKSwmRidGJUYlISIiIiIjRiUsKiokKUYnIiIkRiVGJSokKUYnRi1GJUYsRidGLEYxRixGJSksJkYyRiVGMUYlIiInRiwsJkYnRiVGJUYlRixGLA==

C0m3sol := -512*Zm^5*(Zm-1)^2*(Zm^3-Zm^2-Zm-3)/((Zm^2+3)^6*(Zm+1));

LCQqLiIkNyYiIiIpSSNabUc2IiIiJkYlKSwmRidGJUYlISIiIiIjRiUsKiokKUYnIiIkRiVGJSokKUYnRi1GJUYsRidGLEYxRixGJSksJkYyRiVGMUYlIiInRiwsJkYnRiVGJUYlRixGLA==

n:=12: series(subs(Zm=Zmser(n),C0m3sol),t,n+1);

Ky9JInRHNiIiIiIiIiUiIz4iIiciJCNIIiIpIiVUVSIjNSImdDAnIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlIiM5

algC0m3:=factor(subs(t=sqrt(s),factor(resultant(numer(C0m3-C0m3sol),algZm,Zm)))):

de0:=algeqtodiffeq(algC0m3,C0m3(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUMwbTNHNiI2IyIiIUYn

rec:=diffeqtorec(de0,C0m3(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PC8vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGLC8tRiU2IyIiJCIjPi8tRiU2IyIiJSIkI0gvLUYlNiMiIiYiJVRVLy1GJTYjIiInIiZ0MCcvLUYlNiMiIigiJ1tDJykvLUYlNiMiIikiKStmSTcvLUYlNiMiIioiKkdrS3ciLy1GJTYjIiM1IitOcCcqUUQvLUYlNiMiIzYiLGBNIlF1Ty8tRiU2IyIjNyItR2QxZ1Zg

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

LCYqOiMiJGciIiRWIyIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRictRi02IyNGK0YpISIiLCYqJiIiJUYnLUkjY29HRjE2I0YnRidGJyomIiM6RictRj42IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGSUYnI0YpRitGJ0YnLUYtNiMsJkZJRicjIiIoRjVGJ0YnLUYtNiMsJkZJRicjRlJGKUYnRjktRi02IywmRklGJ0YpRidGOSwmKiYjRikiIzVGJ0ZJRidGJyNGKUY0RidGJ0YnKjgjIiMhKSIjIilGJ0Y2RicsJkY9RicqJkY1RidGQkYnRjlGJ0YsRjlGRUY5RkdGJ0ZKRictRi02IywmRklGJ0YzRidGJ0ZXRjktRi02IywmRklGJyMiIilGKUYnRjksJkZJRidGK0YnRidGOQ==

In fact there seems to be a shorter recurrence relation

n:=180: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C0m3sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 1}));

hypergeomsols(rec1,co(k),{});

LCYqOiMiJGciIiRWIyIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRictRi02IyNGK0YpISIiLCYqJiIjOkYnLUkjY29HRjE2IyIiIUYnRicqJiIiJUYnLUY+NiNGJ0YnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGSUYnIyIiKEY1RidGJy1GLTYjLCZGSUYnI0YpRitGJ0YnLUYtNiMsJkZJRicjRk5GKUYnRjktRi02IywmRklGJ0YpRidGOSwmKiYjRikiIzVGJ0ZJRidGJyNGKUY0RidGJ0Y5KjgjIiMhKSIjIilGJ0Y2RicsJiomRjVGJ0Y9RidGJ0ZDRjlGJ0ZFRjlGLEY5RkdGJ0ZPRictRi02IywmRklGJ0YzRidGJy1GLTYjLCZGSUYnIyIiKUYpRidGOUZXRjksJkZJRidGK0YnRidGJw==

(hypergeomsols(rec1,co(k),{co(2) = 1,co(3) = 19}));

LCYqOCMiI2MiIyIpIiIiKUkjUGlHJSpwcm90ZWN0ZWRHI0YnIiIjISIiKSIiJEYrRictSSZHQU1NQUc2JEYqSShfc3lzbGliRzYiNiMjRixGL0YtLUYxNiMjIiImIiInRicpIiM7SSJrR0Y0RictRjE2IywmRj5GJyMiIihGO0YnRictRjE2IywmRj5GJyNGL0YsRidGJy1GMTYjLCZGPkYnI0ZDRi9GJ0YtLUYxNiMsJkY+RidGL0YnRi0sJiomI0YvIiM1RidGPkYnRicjRi9GOkYnRidGJyo2IyIkLyQiJHYnRidGKEYtRjdGLUYwRidGPEYnRkRGJy1GMTYjLCZGPkYnRjlGJ0YnLUYxNiMsJkY+RicjIiIpRi9GJ0YtRkxGLSwmRj5GJ0YsRidGJ0Yt

seq(%,k=2..10);

NisiIiIiIz4jIiZ0eSMiJCsiIyInLE0oKiIkXSMjIihQKFs4IiNEIyIoNDFdKCIjNSMiKlI7KzAiRjAjIitZekEuUEYtIyItZUlIJD5qIyIkRCI=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-1): no</Text-field>

C1m1ftsol;

LCQqLCIiIyIiIiwuKiQpSSNabUc2IiIiJkYlRiUqJiIjOEYlKUYpIiIlRiUhIiIqJiIjOUYlKUYpIiIkRiVGJSomIiM9RiUpRilGJEYlRiUqJiIiKkYlRilGJUYlRjRGJUYlKSwmRilGJUYlRjBGJEYlLCZGKUYlRiVGJUYwKSwmKiRGN0YlRiVGNEYlRjRGMEYl

C1m1ftsol := (2*(Zm^5-13*Zm^4+14*Zm^3+18*Zm^2+9*Zm+3))*(Zm-1)^2/((Zm+1)*(Zm^2+3)^3);

LCQqLCIiIyIiIiwuKiQpSSNabUc2IiIiJkYlRiUqJiIjOEYlKUYpIiIlRiUhIiIqJiIjOUYlKUYpIiIkRiVGJSomIiM9RiUpRilGJEYlRiUqJiIiKkYlRilGJUYlRjRGJUYlKSwmRilGJUYlRjBGJEYlLCZGKUYlRiVGJUYwKSwmKiRGN0YlRiVGNEYlRjRGMEYl

n:=12: series(subs(Zm=Zmser(n),C1m1ftsol),t,n+1);

Ky9JInRHNiIiIiMiIiUiI0giIiciJCFRIiIpIiVrXCIjNSImO2QnIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

algC1m1ft:=factor(subs(t=sqrt(s),factor(resultant(numer(C1m1ft-C1m1ftsol),algZm,Zm)))):

de0:=algeqtodiffeq(algC1m1ft,C1m1ft(s)): nops(%);

IiIj

op(2,de0);

Ly1JJ0MxbTFmdEc2IjYjIiIhRic=

de:=de0:

rec:=diffeqtorec(de,C1m1ft(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiJkkjX0NHRiZGJy8tRiU2IyIiIywmRjJGLComIyIjWSIiKEYsRi1GLEYsLy1GJTYjIiIkLCYiI0hGLComIiNlRixGLUYsRiwvLUYlNiMiIiUsJiIkIVFGLComIyIlY1RGN0YsRi1GLEYsLy1GJTYjIiImLCYiJWtcRiwqJiMiJkBsJUY3RixGLUYsRiwvLUYlNiMiIicsJiImO2QnRiwqJiMiJ0dHYkY3RixGLUYsRiwvLUYlNiNGNywmIidNTykpRiwqJiMiKCZwY29GN0YsRi1GLEYsLy1GJTYjIiIpLCYiKWdnMDdGLComIyIpLyc+eSlGN0YsRi1GLEYsLy1GJTYjIiIqLCYiKl0lR207RiwqJiMiK100RGA2RjdGLEYtRixGLA==

subs(_C[0]=0,%);

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGMC8tRiU2IyIiJCIjSC8tRiU2IyIiJSIkIVEvLUYlNiMiIiYiJWtcLy1GJTYjIiInIiY7ZCcvLUYlNiMiIigiJ01PKSkvLUYlNiMiIikiKWdnMDcvLUYlNiMiIioiKl0lR207

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmKiZGK0YnLUkjY29HRjE2IyIiIUYnRictRj02I0YnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUY5KjYjRitGJkYnRjZGJywmKiZGTkYnRjxGJ0YnKiZGK0YnRkBGJ0Y5RidGLEY5RkJGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRictRi02IywmRkZGJyNGNEYpRidGOUZTRjlGJw==

In fact there seems to be a shorter recurrence relation

n:=140: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C1m1ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 2, co(3) = 29, co(4) = 380}));

hypergeomsols(rec1,co(k),{});

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnIyIiKEY1RidGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRidGK0YnRjktRi02IywmRkZGJyNGJUYpRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGS0YnRj9GJ0Y5RidGQkY5RixGOUZERictRi02IywmRkZGJ0YzRidGJ0ZMRidGT0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

(hypergeomsols(rec1,co(k),{co(3) = 29, co(4) = 380}));

LCYqNiMiJG0mIiUySCIiIilJI1BpRyUqcHJvdGVjdGVkRyNGJyIiIyEiIikiIiRGK0YnLUkmR0FNTUFHNiRGKkkoX3N5c2xpYkc2IjYjI0YsRi9GLS1GMTYjIyIiJiIiJ0YnKSIjO0kia0dGNEYnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RidGK0YnRictRjE2IywmRj5GJ0YsRidGLS1GMTYjLCZGPkYnIyIiJUYvRidGLUYnKjQjIiQhKSkiJSxFRidGKEYtRjdGLUYwRidGPEYnLUYxNiMsJkY+RidGOUYnRidGREYnRkdGLS1GMTYjLCZGPkYnI0Y6Ri9GJ0YtRi0=

seq(%,k=2..10);

NisjIiZVNSIiJSJcJiIjSCIkIVEjIigmNHNHIiR5JiMiKiFmYTtPRiUjIitOT3JQKCoiJiMpNCIjIiwpcDtqXW1GJSMiLDxNSEdUJiIkQiQjIiwpKVxMZylSIiM8

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (2,-1): no</Text-field>

C2m1sol ;

C2m1sol := -32*Zm^3*(Zm-1)^2*(Zm^8-32*Zm^7+166*Zm^6-68*Zm^5-212*Zm^4-200*Zm^3-94*Zm^2+44*Zm+11)/((Zm+1)^2*(Zm^2+3)^6);

n:=12: series(subs(Zm=Zmser(n),C2m1sol),t,n+1);

Ky9JInRHNiIiIiQiIiUiI2ciIiciJFwqIiIpIiZQUyIiIzUiJzJJPyIjNy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzk=

algC2m1:=factor(subs(t=sqrt(s),factor(resultant(numer(C2m1-C2m1sol),algZm,Zm)))):

de0:=algeqtodiffeq(algC2m1,C2m1(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUMybTFHNiI2IyIiIUYn

de:=de0:

rec:=diffeqtorec(de,C2m1(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PDAvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMiIiQvLUYlNiNGMSIjZy8tRiU2IyIiJSIkXCovLUYlNiMiIiYiJlBTIi8tRiU2IyIiJyInMkk/Ly1GJTYjIiIoIig+bSJILy1GJTYjIiIpIikkNCkqPSUvLUYlNiMiIioiKlIpeU1nLy1GJTYjIiM1Iit1aVlEKCkvLUYlNiMiIzYiLXFVLSRwRSIvLUYlNiMiIzciLmRzOGl1JT0vLUYlNiMiIzgiL0t3JzRAX3Ej

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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

In fact there seems to be a shorter recurrence relation

n:=220: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C2m1sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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rec1:=map(factor,op(1,rec));

LCwqNiIlJzQlIiIiLCYqJiIiJ0YlSSJrRzYiRiVGJSIiJkYlRiUsJiomIiIkRiVGKUYlRiVGK0YlRiUsJiomIiIjRiVGKUYlRiVGLkYlRiUsJkYwRiVGJUYlRiUsJkYtRiUiIihGJUYlLCZGJ0YlRjRGJUYlLCZGKUYlRiVGJUYlLDgqJiIoK3c8J0YlKUYpIiM1RiVGJSomIipvZm4lPUYlKUYpIiIqRiVGJSomIitrVGImXCNGJSlGKSIiKUYlRiUqJiIsYFJ1IT0/RiUpRilGNEYlRiUqJiItQ1E0eiczIkYlKUYpRihGJUYlKiYiLTQkcDJoMyVGJSlGKUYrRiVGJSomIi4iKj0zSnIzIkYlKUYpIiIlRiVGJSomIi5LQ1lpcCwjRiUpRilGLkYlRiUqJiIuLCc9U10nWyNGJSlGKUYxRiVGJSomIi5lSWlCKUc9RiVGKUYlRiUiLVNJbjRlZ0YlRiUtSSNjb0dGKjYjRilGJUYlKioiJEciRiVGL0YlLEQqJiIsKyV5WS1LRiUpRikiIztGJUYlKiYiLjcoPlNrRDhGJSlGKSIjOkYlRiUqJiIvI1IkR0MhKmZERiUpRikiIzlGJUYlKiYiME9SVSg+VGxJRiUpRikiIzhGJUYlKiYiMU9ScEhaPl5ERiUpRikiIzdGJUYlKiYiMk9GTEc4WmxjIkYlKUYpIiM2RiVGJSomIjJjcSFmWlVaWnRGJUY6RiVGJSomIjN5c0NSblUkZW8jRiVGPkYlRiUqJiIzUUN1YF4vUkl4RiVGQkYlRiUqJiI0MlArIXl4J3pldiJGJUZGRiVGJSomIjQjKlEkWzFsXGtKSkYlRklGJUYlKiYiNDZOd0ZheF4nSFZGJUZMRiVGJSomIjQsdHoyb0Nwa2AlRiVGT0YlRiUqJiI0cVtBcig0ZWtyTUYlRlNGJUYlKiYiNCYpejRIKUdYK0M9RiVGVkYlRiUqJiIzXTEkKVFiMztlZUYlRilGJUYlIjIrPyVbSC1kTCcpRiVGJS1GZW42I0Y2RiUhIiIqKiIjW0YlLCZGKUYlRlBGJUYlLEQqJiIsKyNSQiw7RiVGXG9GJUYlKiYiLWMnPTAlb29GJUZgb0YlRiUqJiIvVyo9M0JBUCJGJUZkb0YlRiUqJiIwbzpjVUR2cCJGJUZob0YlRiUqJiIxV2wlSC9YeVgiRiVGXHBGJUYlKiYiMV9sMzd4UkwjKkYlRmBwRiVGJSomIjJzNjsmZlpHcFdGJUY6RiVGJSomIjNSImZmKHkjcCkpbyJGJUY+RiVGJSomIjMtMCdIUCZSU1JdRiVGQkYlRiUqJiI0LklUJ1JfPEMiPiJGJUZGRiVGJSomIjRmJlFOVzRzKTNBI0YlRklGJUYlKiYiND1BXlM+XSI+Q0tGJUZMRiVGJSomIjR6RDEpWyl6bj9jJEYlRk9GJUYlKiYiNGtnYzhyI0g5JSlHRiVGU0YlRiUqJiI0KzwhZW5XKj1zZyJGJUZWRiVGJSomIjMrZyhRanN6SVsmRiVGKUYlRiUiMis/LkpHZyopZSlGJUYlLUZlbjYjLCZGKUYlRjFGJUYlRiUqMEZDRiUsJkYtRiVGZW9GJUYlLCZGLUYlRmlvRiVGJSwmRilGJUYrRiVGJUZcckYlLD4qJiIqK3N5VyVGJUZob0YlRiUqJiIsJ0hKNWo5RiVGXHBGJUYlKiYiLW9WUDUrQUYlRmBwRiVGJSomIi5jUy9nPCwjRiVGOkYlRiUqJiIvUXFLcSRcRCJGJUY+RiVGJSomIi9mJT5dWTluJkYlRkJGJUYlKiYiMFRWaiNcJyk9PkYlRkZGJUYlKiYiMDcpUkopXCVIXEYlRklGJUYlKiYiMDJoa0tKYmYqRiVGTEYlRiUqJiIxXUgmM1Z6KipRIkYlRk9GJUYlKiYiMXcxKHBAXllXIkYlRlNGJUYlKiYiMUZoaC87KzY1RiVGVkYlRiUqJiIwcSZcdlpDTFVGJUYpRiVGJSIvKycqPWMpKlx6RiVGJS1GZW42IywmRilGJUYuRiVGJUZpcSo0LCZGLUYlIiM8RiVGJUZjdEYlLCZGLUYlRl1vRiVGJUZkdEYlLCZGKUYlRihGJUYlRmV0RiVGXHJGJSw4RjhGJSomIipvKioqRzdGJUY+RiVGJSomIitfQ1k2NkYlRkJGJUYlKiYiKypbTEw3J0YlRkZGJUYlKiYiLEx1IlIyQkYlRklGJUYlKiYiLGlIQCRIaUYlRkxGJUYlKiYiLWopPlhsPyJGJUZPRiVGJSomIi1oZlU1QjtGJUZTRiVGJSomIi1fd1EiPlUiRiVGVkYlRiUqJiIsZzcsZj8oRiVGKUYlRiUiLCtXS2plIkYlRiUtRmVuNiNGXHJGJUYl

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 3, co(3) = 60}));

hypergeomsols(rec1,co(k),{});

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(hypergeomsols(rec1,co(k),{co(2) = 3, co(3) = 60}));

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seq(%,k=2..10);

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<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-2): no</Text-field>

C1m2sol;

C1m2sol := 32*Zm^3*(Zm-1)^2*(Zm^7-9*Zm^6-Zm^5-19*Zm^4+79*Zm^3+33*Zm^2+81*Zm+27)/((Zm^2+3)^6*(Zm+1));

n:=12: series(subs(Zm=Zmser(n),C1m2sol),t,n+1);

Ky9JInRHNiIiIiQiIiUiI2AiIiciJCF5IiIpIiYxNSIiIzUiJ3BSOiIjNy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzk=

algC1m2:=factor(subs(t=sqrt(s),factor(resultant(numer(C1m2-C1m2sol),algZm,Zm)))):

de0:=algeqtodiffeq(algC1m2,C1m2(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUMxbTJHNiI2IyIiIUYn

de:=de0:

rec:=diffeqtorec(de,C1m2(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

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Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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In fact there seems to be a shorter recurrence relation

n:=220: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C1m2sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 3, co(3) = 53}));

hypergeomsols(rec1,co(k),{});

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

(hypergeomsols(rec1,co(k),{co(2) = 3, co(3) = 53}));

LCYqOCMiJCRmIiUuXiIiIilJI1BpRyUqcHJvdGVjdGVkRyNGJyIiIyEiIikiIiRGK0YnLUkmR0FNTUFHNiRGKkkoX3N5c2xpYkc2IjYjI0YsRi9GLS1GMTYjIyIiJiIiJ0YnKSIjO0kia0dGNEYnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RidGK0YnRictRjE2IywmRj5GJyNGQ0YvRidGLS1GMTYjLCZGPkYnRi9GJ0YtLCgqJClGPkYsRidGJyomIiM1RidGPkYnRidGPUYnRidGJyo2IyIlLTpGJkYnRihGLUY3Ri1GMEYnRjxGJy1GMTYjLCZGPkYnRjlGJ0YnRkRGJy1GMTYjLCZGPkYnIyIiKUYvRidGLUZLRi0sKEZPRicqJkZnbkYnRj5GJ0YtIiM/Ri1GJ0Yn

seq(%,k=2..10);

NisiIiQiI2AjIiYoUSYpIiRFIiMiJ0xucSIjJSkjIig0Iz5BIiNAIyIpKHAhKXkkIiNHIyIrbz85NzYiI2ojIixqIz0jZSFmIiRfIyMiLFVTSkklRyIiKg==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0): no</Text-field>

C00ftsol;

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

C00ftsol:=-(Zm-1)*(Zm+1)*(Zm^4-6*Zm^2-11)/(Zm^2+3)^3;

LCQqKiwmSSNabUc2IiIiIkYnISIiRicsJkYlRidGJ0YnRicsKCokKUYlIiIlRidGJyomIiInRicpRiUiIiNGJ0YoIiM2RihGJyksJiokRjBGJ0YnIiIkRidGNkYoRig=

algC00ft:=subs(t=sqrt(s),factor(resultant(numer(C00ft-C00ftsol),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algC00ft,C00ft(s)));

LDAqJiIlWz8iIiIpSSJzRzYiIiIkRiUhIiIqJiIkYyNGJSlGJyIiI0YlRiUqJiIjU0YlRidGJUYqKihGLkYlLCoqJiIlP15GJUYmRiVGJSomIiU/PkYlRi1GJUYqKiYiJEsiRiVGJ0YlRiVGJUYqRiUtSSZDMDBmdEdGKDYjRidGJUYqKipGLkYlRidGJSwsKiYiJ2dgbEYlKUYnIiIlRiVGJSomIiYhb3JGJUYmRiVGJSomIiVzSUYlRi1GJUYqKiYiJCVlRiVGJ0YlRioiIz5GJUYlLUklZGlmZkclKnByb3RlY3RlZEc2JEY5RidGJUYlKi4iIipGJUYtRiUsJiomIiM7RiVGJ0YlRiVGJUYqRiUsJkZQRiVGJUYlRiUsKComIiVDNUYlRi1GJUYlKiYiI1tGJUYnRiVGJSIiJkYqRiUtRko2JEZJRidGJUYlKixGTkYlRiZGJSlGT0YuRiUpRlJGLkYlLUZKNiRGWUYnRiVGJQ==

rec:=diffeqtorec(de,C00ft(s),co(k)):

rec1:=map(factor,op(1,rec));

LCwqLCImT2InIiIiSSJrRzYiRiUsJiomIiIkRiVGJkYlRiUiIiNGJUYlLCZGKUYlRiVGJUYlLUkjY29HRic2I0YmRiVGJSooIiVbP0YlLCgqJiIjYUYlKUYmRitGJUYlKiYiJEMiRiVGJkYlRiUiI2xGJUYlLUYuNiMsJkYmRiVGJUYlRiVGJSoqIiRvKEYlLCYqJkYrRiVGJkYlRiUiIiZGJUYlLCgqJkYqRiVGNUYlRiUqJiIjOkYlRiZGJUYlIiM4RiVGJS1GLjYjLCZGJkYlRitGJUYlISIiKigiIilGJSwoRjNGJSomIiQ7JUYlRiZGJUYlIiQmekYlRiUtRi42IywmRiZGJUYqRiVGJUZJKiosJkYmRiVGQEYlRiUsJkYpRiUiIzlGJUYlLCZGKUYlRkVGJUYlLUYuNiMsJkYmRiUiIiVGJUYlRiU=

{seq(op(i,rec),i=2..nops(rec))};

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIigvLUYlNiMiIiQiI2w=

No HG solution

(hypergeomsols(rec1,co(k),%));

(hypergeomsols(rec1,co(k),{}));

IiIh

In fact there seems to be a shorter recurrence relation

n:=70: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C00ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

PCUsKComLCwqJiIlOyMqIiIiKUkia0c2IiIiJkYoRigqJiImITNZRigpRioiIiVGKEYoKiYiJiEzeUYoKUYqIiIkRihGKComIiZndCVGKClGKiIiI0YoRigqJiIlLygpRihGKkYoRihGKC1JI2NvR0YrNiNGKkYoRigqJiwuKiYiJV82RihGKUYoISIiKiYiJVMnKUYoRi9GKEZCKiYiJitbI0YoRjNGKEZCKiYiJlNRJEYoRjdGKEZCKiYiJjNBI0YoRipGKEZCIiUhKWVGQkYoLUY8NiMsJkYqRihGKEYoRihGKComLC4qJiIjT0YoRilGKEYoKiYiJGckRihGL0YoRigqJiIlJlEiRihGM0YoRigqJiIlZ0RGKEY3RihGKComIiUqSCNGKEYqRihGKCIkUylGKEYoLUY8NiMsJkYqRihGOEYoRihGKC8tRjw2IyIiIUZcby8tRjw2I0YoRig=

rec1:=map(factor,op(1,rec));

LCgqLiIkYyMiIiJJImtHNiJGJSwmKiYiIiRGJUYmRiVGJSIiI0YlRiUsJkYpRiVGJUYlRiUsKComIiIlRiUpRiZGK0YlRiUqJiIjO0YlRiZGJUYlIiM8RiVGJS1JI2NvR0YnNiNGJkYlRiUqKiIiKUYlLCYqJkYrRiVGJkYlRiVGKkYlRiUsLComIiNzRiUpRiZGL0YlRiUqJiIkSyVGJSlGJkYqRiVGJSomIiQtKkYlRjBGJUYlKiYiJGkoRiVGJkYlRiUiJFgjRiVGJS1GNTYjLCZGJkYlRiVGJUYlISIiKiwsJkYpRiUiIihGJUYlLCZGJkYlRipGJUYlLCZGKUYlRjhGJUYlLChGLkYlKiZGOEYlRiZGJUYlIiImRiVGJS1GNTYjLCZGJkYlRitGJUYlRiU=

(hypergeomsols(rec1,co(k),{}));

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,0): no</Text-field>

C10sol;

C10sol:=(32*(Zm-1))*Zm^3*(Zm^8-10*Zm^7+8*Zm^6-18*Zm^5+82*Zm^4-14*Zm^3+112*Zm^2-22*Zm-11)/((Zm^2+3)^6*(Zm+1));

algC10:=subs(t=sqrt(s),factor(resultant(numer(C10-C10sol),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algC10,C10(s))):

nops(de);

IiIj

op(2,de);

Ly1JJEMxMEc2IjYjIiIhRic=

rec:=diffeqtorec(de,C10(s),co(k)):

rec1:=map(factor,op(1,rec)):

{seq(op(i,rec),i=2..nops(rec))};

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIzgvLUYlNiMiIiQiJGYiLy1GJTYjIiIlIiUkKT4vLUYlNiMiIiYiJiVSRC8tRiU2IyIiJyInOUlMLy1GJTYjIiIoIig0YlglLy1GJTYjIiIpIik4Ij0xJy8tRiU2IyIiKiIqJXolUk8pLy1GJTYjIiM1IixrUGl5OyIvLUYlNiMiIzYiLSgpKXojUlo7Ly1GJTYjIiM3Ii4wSnJuVk0jLy1GJTYjRjEiL0tfcl4iPk8kLy1GJTYjIiM5IjA6K3kuUFAmWw==

some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

(hypergeomsols(rec1,co(k),{}));

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

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C10sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec));

LCwqNiIlJzQlIiIiLCYqJiIiJ0YlSSJrRzYiRiVGJSIiJkYlRiUsJiomIiIkRiVGKUYlRiVGK0YlRiUsJiomIiIjRiVGKUYlRiVGLkYlRiUsJkYwRiVGJUYlRiUsJkYtRiUiIiVGJUYlLCZGJ0YlIiIoRiVGJSwmRilGJUYlRiVGJSw6KiYiJjNBJ0YlKUYpIiM2RiVGJSomIihzUSdHRiUpRikiIzVGJUYlKiYiKS92eWZGJSlGKSIiKkYlRiUqJiIqK3B2UShGJSlGKSIiKUYlRiUqJiIrT2oqKW9mRiUpRilGNkYlRiUqJiIsaCF6Pi5MRiUpRilGKEYlRiUqJiItdik+KilvRiJGJSlGKUYrRiVGJSomIi0hKnBSQlxNRiUpRilGNEYlRiUqJiItQDhlRydRJ0YlKUYpRi5GJUYlKiYiLWQ9WzFGeEYlKUYpRjFGJUYlKiYiLWM3VjwwYkYlRilGJUYlIi0/Q11tXjxGJUYlLUkjY29HRio2I0YpRiVGJSoqIiRHIkYlRi9GJSxGKiYiKnNpW0EkRiUpRikiIzxGJUYlKiYiLC8kPUNCPUYlKUYpIiM7RiVGJSomIi1fXnVAMVtGJSlGKSIjOkYlRiUqJiIuT05Oay8heUYlKUYpIiM5RiVGJSomIi8lUSVRLGArKClGJSlGKSIjOEYlRiUqJiIwUyU0JDMsWzEoRiUpRikiIzdGJUYlKiYiMV8kKSpmTzAzSyVGJUY7RiVGJSomIjJvZDRoUl9DLiNGJUY/RiVGJSomIjJjQzNNZChIV3VGJUZDRiVGJSomIjNlMT0+S0FpTkBGJUZHRiVGJSomIjM/KCo0JVJvcykqeiVGJUZLRiVGJSomIjMsbSlSZ0wvZlMpRiVGTkYlRiUqJiI0amVmUm0iKVJMOCJGJUZRRiVGJSomIjQpUj4jeTJ0Wj86IkYlRlRGJUYlKiYiMyw0WEchb1YoRyYpRiVGV0YlRiUqJiIzJnBKKj5TRkhNVkYlRlpGJUYlKiYiMytbQSopeiNSL04iRiVGKUYlRiUiMitOSTNAYEclPkYlRiUtRmluNiNGN0YlISIiKioiI1tGJSwmRilGJUY0RiVGJSxGKiYiKk9KQ2giRiVGYG9GJUYlKiYiK2NRMmUkKkYlRmRvRiVGJSomIi0hby48I1FERiVGaG9GJUYlKiYiLiUpZiFIXV5VRiVGXHBGJUYlKiYiL2tTLSQ0JTRcRiVGYHBGJUYlKiYiMFsnPj0lSCRSVEYlRmRwRiVGJSomIjFvXjRUUWxORUYlRjtGJUYlKiYiMiNwIWZhPyJvJEgiRiVGP0YlRiUqJiIyLylvIltaclEmXEYlRkNGJUYlKiYiM0pOekM+NTcpWyJGJUZHRiVGJSomIjNNNDkkKm9YWzFORiVGS0YlRiUqJiIzPjJLUE1mLVdrRiVGTkYlRiUqJiIzcmpeZnV2NEIiKkYlRlFGJUYlKiYiM1MiMyxaO006dSpGJUZURiVGJSomIjNWMlZcMScqb3d2RiVGV0YlRiUqJiIzSWFecCRRSV0vJUYlRlpGJUYlKiYiMysibzIlM2ptQjhGJUYpRiVGJSIyKyE0WCJvPSMqKj5GJUYlLUZpbjYjLCZGKUYlRjFGJUYlRiUqMEZIRiUsJkYtRiVGYXBGJUYlLCZGKUYlRitGJUYlRmJyRiUsJkYtRiVGXXBGJUYlLEAqJiIodyp5V0YlRlxwRiVGJSomIipPbjo6I0YlRmBwRiVGJSomIiszZSRcdyVGJUZkcEYlRiUqJiIsZ2AkUSpRJ0YlRjtGJUYlKiYiLSs5aW1pZEYlRj9GJUYlKiYiLkd5JUgrJ28kRiVGQ0YlRiUqJiIvOXFYQUZAPEYlRkdGJUYlKiYiL2woPXlpZyZmRiVGS0YlRiUqJiIweDR3bWFTYCJGJUZORiVGJSomIjAiKSk+Iyk9ZEVIRiVGUUYlRiUqJiIweXJzYyZcclNGJUZURiVGJSomIjArd1xgSnErJUYlRldGJUYlKiYiMFphQic9aVFFRiVGWkYlRiUqJiIwSUouM0Y4LyJGJUYpRiVGJSIvKyllKDNlZj1GJUYlLUZpbjYjLCZGKUYlRi5GJUYlRl9yKjQsJkYtRiVGZW9GJUYlRlt1RiUsJkYpRiVGKEYlRiVGXHVGJUZickYlLCZGLUYlRmFvRiVGJUZddUYlLDpGOUYlKiYiKCVlekBGJUY/RiVGJSomIilDLWRNRiVGQ0YlRiUqJiIqJUd6Iz4kRiVGR0YlRiUqJiIrIUdiISkpPUYlRktGJUYlKiYiKyhmP2FbKEYlRk5GJUYlKiYiLEA/djguI0YlRlFGJUYlKiYiLFlQdjJ5JEYlRlRGJUYlKiYiLG5pTTl1JUYlRldGJUYlKiYiLCp6eklFUUYlRlpGJUYlKiYiLDUzcUp6IkYlRilGJUYlIisrRUkxUEYlRiUtRmluNiNGYnJGJUYl

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 1, co(2) = 13, co(3) = 159}));

(hypergeomsols(rec1,co(k),{}));

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

(hypergeomsols(rec1,co(k),{co(2) = 13, co(3) = 159}));

LCYqOCMiI2YiJDcnIiIiKUkjUGlHJSpwcm90ZWN0ZWRHI0YnIiIjISIiKSIiJEYrRictSSZHQU1NQUc2JEYqSShfc3lzbGliRzYiNiMjRixGL0YtLUYxNiMjIiImIiInRicpIiM7SSJrR0Y0RictRjE2IywmRj5GJyMiIihGO0YnRictRjE2IywmRj5GJ0YrRidGJy1GMTYjLCZGPkYnI0ZDRi9GJ0YtLUYxNiMsJkY+RidGL0YnRi0sKCokKUY+RixGJ0YnKiYjIiM1Ri9GJ0Y+RidGJyMiIyMpIiNGRidGJ0YtKjYjIiVGJioiJWFGRidGKEYtRjdGLUYwRidGPEYnLUYxNiMsJkY+RidGOUYnRidGREYnLUYxNiMsJkY+RicjIiIpRi9GJ0YtRktGLSwoRk9GJ0Y+RicjRjpGL0YtRidGJw==

seq(%,k=2..10);

NisiIzgiJGYiIyIoTmUkSCIlSzsjIic8OEwiIzsjIipWRVpMIiIkVyYjIitAbnFRQyIkOykjIixERSJSOTUiJHMjIyIrXCEzO2coRiojIi5SYnhZcC0mRjA=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,1): no</Text-field>

C01sol ;

C01sol := -64*Zm^3*(Zm-1)*(9*Zm^6-14*Zm^5+3*Zm^4-44*Zm^3-25*Zm^2-38*Zm-19)/((Zm^2+3)^6*(Zm+1));

LCQqLiIjayIiIilJI1ptRzYiIiIkRiUsJkYnRiVGJSEiIkYlLDAqJiIiKkYlKUYnIiInRiVGJSomIiM5RiUpRiciIiZGJUYrKiZGKUYlKUYnIiIlRiVGJSomIiNXRiVGJkYlRisqJiIjREYlKUYnIiIjRiVGKyomIiNRRiVGJ0YlRisiIz5GK0YlKSwmKiRGPEYlRiVGKUYlRjBGKywmRidGJUYlRiVGK0Yr

algC01:=subs(t=sqrt(s),factor(resultant(numer(C01-C01sol),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algC01,C01(s))):

nops(de);

IiIj

op(2,de);

Ly1JJEMwMUc2IjYjIiIhRic=

rec:=diffeqtorec(de,C01(s),co(k)):

rec1:=map(factor,op(1,rec)):

{seq(op(i,rec),i=2..nops(rec))};

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIiIjLy1GJTYjRi0iI0AvLUYlNiMiIiQiJEsjLy1GJTYjIiIlIiVBRi8tRiU2IyIiJiImWU0kLy1GJTYjIiInIiciZkQlLy1GJTYjIiIoIihyVGMmLy1GJTYjIiIpIikmNERWKC8tRiU2IyIiKiIrIm9iLSwiLy1GJTYjIiM1IiwpUSFvSVIiLy1GJTYjIiM2Ii1dJzRqVSU+Ly1GJTYjIiM3Ii5mMDFnOnUjLy1GJTYjIiM4Ii9zKSplQDkrUi8tRiU2IyIjOSIwNTZBUXc2ZiY=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

(hypergeomsols(rec1,co(k),{}));

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

In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C01sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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

rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 2, co(2) = 21, co(3) = 232}));

(hypergeomsols(rec1,co(k),{}));

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

(hypergeomsols(rec1,co(k),{co(2) = 21, co(3) = 232}));

LCYqOCMiJS86IiRiKSIiIilJI1BpRyUqcHJvdGVjdGVkRyNGJyIiIyEiIikiIiRGK0YnLUkmR0FNTUFHNiRGKkkoX3N5c2xpYkc2IjYjI0YsRi9GLS1GMTYjIyIiJiIiJ0YnKSIjO0kia0dGNEYnLUYxNiMsJkY+RidGK0YnRictRjE2IywmRj5GJyMiIihGO0YnRictRjE2IywmRj5GJyNGRkYvRidGLS1GMTYjLCZGPkYnRi9GJ0YtLCgqJClGPkYsRidGJyomIyIjNUYvRidGPkYnRicjIiNpIiNGRidGJ0YnKjYjIidvbzciJmpoI0YnRjdGLUYoRi1GMEYnRjxGJy1GMTYjLCZGPkYnRjlGJ0YnRj9GJy1GMTYjLCZGPkYnIyIiKUYvRidGLUZLRi0sKEZPRicqJiMiI1oiI0JGJ0Y+RidGJyMiIz8iI3BGJ0YnRi0=

seq(%,k=2..10);

NisiI0AiJEsjIyIoc2x2IyIkcCojIiljaCM9IiIkQiQjIiptUiJvOkYqIyIsJWVSLic9JCIlWFsjIi1PYF1DaTkiJTo7IyIrUHktO2oiIiYjIi1oOWVvc0UiIzo=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,1): no</Text-field>

C11ftsol;

KiopLCZJI1ptRzYiIiIiRichIiIiIiNGJyksJkYlRidGJ0YnRilGJywsKiQpRiUiIilGJ0YnKiYiI1dGJylGJSIiJ0YnRigqJiIkNSJGJylGJSIiJUYnRicqJiIkPyVGJylGJUYpRidGJyIkIkdGJ0YnKSwmKiRGOkYnRiciIiRGJ0YzRig=

C11ftsol := (Zm-1)^2*(Zm+1)^2*(Zm^8-44*Zm^6+110*Zm^4+420*Zm^2+281)/(Zm^2+3)^6;

KiopLCZJI1ptRzYiIiIiRichIiIiIiNGJyksJkYlRidGJ0YnRilGJywsKiQpRiUiIilGJ0YnKiYiI1dGJylGJSIiJ0YnRigqJiIkNSJGJylGJSIiJUYnRicqJiIkPyVGJylGJUYpRidGJyIkIkdGJ0YnKSwmKiRGOkYnRiciIiRGJ0YzRig=

algC11ft:=subs(t=sqrt(s),factor(resultant(numer(C11ft-C11ftsol),algZm,Zm))):

de:=map(factor,algeqtodiffeq(algC11ft,C11ft(s)));

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

rec:=diffeqtorec(de,C11ft(s),co(k)):

rec1:=map(factor,op(1,rec));

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

{seq(op(i,rec),i=2..nops(rec))};

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMiIiQvLUYlNiNGMSIjVy8tRiU2IyIiJSIkImUvLUYlNiMiIiYiJUd3Ly1GJTYjIiInIic+ODU=

No HG solution

(hypergeomsols(rec1,co(k),%));

(hypergeomsols(rec1,co(k),{}));

IiIh

In fact there seems to be a shorter recurrence relation

n:=70: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),C11ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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rec1:=map(factor,op(1,rec));

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

(hypergeomsols(rec1,co(k),{}));

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,1) : no</Text-field>

Cm21sol:=-(64*(Zm-1))*(Zm^5+Zm^4+2*Zm^3-30*Zm^2-19*Zm-19)*Zm^3/(Zm^2+3)^6;

LCQqLCIjayIiIiwmSSNabUc2IkYlRiUhIiJGJSwuKiQpRiciIiZGJUYlKiQpRiciIiVGJUYlKiYiIiNGJSlGJyIiJEYlRiUqJiIjSUYlKUYnRjJGJUYpKiYiIz5GJUYnRiVGKUY5RilGJUYzRiUpLCYqJEY3RiVGJUY0RiUiIidGKUYp

n:=12: series(subs(Zm=Zmser(n),Cm21sol),t,n+1);

Ky9JInRHNiIiIiNGJSIjPyIiJSIkOCMiIiciJUlDIiIpIiYwI0giIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

algCm21:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm21-Cm21sol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm21,Cm21(s)): nops(%);

IiIj

op(2,de0);

Ly1JJUNtMjFHNiI2IyIiIS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqJClJI19aR0YqIiIjIiIiRjNGMSEiIg==

de:={op(1,de0),Cm21(0)=0}: :

rec:=diffeqtorec(de,Cm21(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIiIjLy1GJTYjRi0iIz8vLUYlNiMiIiQiJDgjLy1GJTYjIiIlIiVJQy8tRiU2IyIiJiImMCNILy1GJTYjIiInIic9XU8vLUYlNiMiIigiKEI8cSUvLUYlNiMiIikiKSY+Pj8nLy1GJTYjIiIqIipgI0hSJCkvLUYlNiMiIzUiLGBNciJSNi8tRiU2IyIjNiItKD4nXCNvZCIvLUYlNiMiIzciLkpbKioqPjJBLy1GJTYjIiM4Ii9bKG9WSCU+Si8tRiU2IyIjOSIwYGNCJEdnWFc=

Some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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In fact there seems to be a shorter recurrence relation

n:=200: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm21sol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

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rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 2, co(2) = 20, co(3) = 213}));

hypergeomsols(rec1,co(k),{});

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(hypergeomsols(rec1,co(k),{co(2) = 20, co(3) = 213}));

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seq(%,k=2..10);

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<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,2): no</Text-field>

Cm22ftsol;

Cm22ftsol:=16*(Zm-1)^2*(Zm^8-6*Zm^7+2*Zm^6-30*Zm^5+80*Zm^4+38*Zm^3+110*Zm^2-2*Zm-1)/(Zm^2+3)^6;

n:=12: series(subs(Zm=Zmser(n),Cm22ftsol),t,n+1);

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algCm22ft:=factor(subs(t=sqrt(s),factor(resultant(numer(Cm22ft-Cm22ftsol),algZm,Zm)))):

de0:=algeqtodiffeq(algCm22ft,Cm22ft(s)): nops(%);

IiIj

op(2,de0);

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de:={op(1,de0),Cm22ft(0)=0} :

rec:=diffeqtorec(de,Cm22ft(s),co(k)):

rec1:=map(factor,op(1,%)):

{seq(op(i,rec),i=2..nops(rec))};

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some HG solutions, but not ours

(hypergeomsols(rec1,co(k),%));

hypergeomsols(rec1,co(k),{});

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In fact there seems to be a shorter recurrence relation

n:=220: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),Cm22ftsol),s,n/2+1)):

rec:=op(1,seriestorec(ser,co(k)));

<),0*&,8*&"drcY$\_\R*QJFUm%=%*)Q+bw@[ZdqULjoY()HKQ07fG$3n?@TtCI3_>!>PmJ>h"4I0Kh""")I"kG6""#6F(!""*&"fr!3qE'z2<#*f_'*3)H\@y]"\P3wB0O'\z;"4R)H:G+CaQad/$Qu/(4i%))H$fM_#ePf902-"F()F*"#5F(F-*&"frS564U%o&*RpG3^))=!)zV(Royt+!f*Ho*)*QxKaDz(Qzew/U-S/D%Qe;BK`N@bpZx<qqqF()F*""*F(F-*&"gr!GH<Ui!)oYLc7/f0"f2<Q6O\T\t6ur)z&pAmX'=f2f0>=^'y#Q&Q&p[:JS.f$Rtbic#p#F()F*"")F(F-*&"gr[%GNmMD"*)4[$y]bnwU<Kp_Y'*ypei5`Va42_txsFc*eT;SwuIXfS-T(>&3Sg>/2=oiF()F*""(F(F-*&"grgLI[./W//f)=F(Q2ka2/!3\1'o1i:w!p*>Y:&*p?'RgAQ0y(H,O(zKD#=X")[u.MH$ejN*F()F*""'F(F-*&"grg@Ds@#puNVD!=KpS3`nj]\vz[4p[p*p7C1_tL%*eN!\3DE*=h2kDvtANvL'458vA%4*F()F*""&F(F-*&"gr?ZT_(o$>#or=&zgZMWusXC#f8u8]%H8gy))=*z%)>RszdV)3-;$y$z1$QY]fUHLQU^BIp&F()F*""%F(F-*&"gr'H:#o=6KmYlmPmw'*4$z#\.HEcEm6$fFg1GqG"y(QJ&pv'RN*[s"*R!3[?"e*e=!QIZ:(z>#F()F*""$F(F-*&"frg\10vn5ItPKjY$eu+]X^.KFE+LV$)yc[rT%G%H'>&*e%3KkWxMTlS9$e!z!\b"QB20CZF()F*""#F(F-*&"er+C#3^KBRaUrN/Xo(of/y3&)eA"fi!*\%[L"G$4z)49Z(on7tf#e)\^5Rn!zmxe#[Z#HhG%F(F*F(F-F(-I#coGF+6#F*F(F(*&,:*&"drci4)GK['Qq1x\W@O%Hk)=j*pPO-5G<L*e#z<+8LH'3E`kX$\$QxZT(ot(*H!HBXp!)*4AF(F)F(F(*&"er?f=BmTbPF)z))y(zz@5n&Q)f)))4^h$R5CM#*Ra279HVAwy]2"HBND@[-*G8c[](RJ["\F(F0F(F(*&"fr+)3Dgh#3+-'>!Rx4#zO!Ge4feKR"H#3&))41G4U"*o8![)*3+0'3$pMVQBo$y!\5#*GcvH`k%F(F4F(F(*&"gr+Kw(yK1kaqEQ,>GjeAEs^nF@vV"))*\(RT5rCY]MS_19f&>$R9<8569%RvfzqL&=a%R#F(F8F(F(*&"gr3ys?'4n&p#\#*p0nPv3a)41;\/OGpC"*>Mk6&f6o!)*oYk@Uk]I8Rep)=]s@$***z*[EnspF(F<F(F(*&"grg\'=qE_4S$3cMDlFz()z96U%>gO-$eSELEs-Oli)z1$>E)zYB`OpZ3&)y/D81l6e(>*=*F(F@F(F(*&"gr+%)*p<sP4<T<"e\>tjg7'GsRGJ-!>?4Rn]b\/(e,Q!Hb!\Z)p7g#[,PYA3B@:QSQ#4%=)F(FDF(F-*&"hr+[#*ep$=nNwbm!=nZt+8#='HP@lcy2s*fcYGM@Rqp)*)3hS))zv$=:LLv!*\'*p4[tg/$zp&F(FHF(F-*&"irk_V>p^*[rC%43=3?g5kv>W#)="*>9_=r8e'o5&H[dyG-.zxH=5PX"RYcko,4uH&*Gck2"F(FLF(F-*&"ir!3#yW9!GTuj0#zEI%f:q3,9IbiRoT=H.b:zr+"o-<^@?I6(H,P()HWwTt8M_osH(els5F(FPF(F-*&"hr+oz1Us]I,#RlE()H;8;H0&y?Nm;!\Wyo3!4O23uIQ;Tb"\-ra4bq+8rLvl&e#*HCU$)o&F(F*F(F-"hr+g$R-=O&f*ywiPd-5D"fnbcY#\<Be+e7Em()f&)GC>l&yFz4KP?R>.,-=1RN$*oz&)4iE"F-F(-FU6#,&F*F(F(F(F(F(*&,:*&"crg<U=O(*Q%Q3^5aed"y&)G9oS&QHx"z\)>r[NQ-@KdtMw#*R)R9/x(H">C>n^+@%\R:2$F(F)F(F-*&"dr+G`"4&G$o3q1m*ys,WpG'Q/*eo#ef"fm!>kpr0nN1m#GE!)G$>prs%)[!47E&HJ$[20S.)F(F0F(F-*&"er!3Q'=**yE+'*=*4>3=?$)3'RAl6$z\J]+USRyC$)[Cx%[`qe@!*)\Ui!*o>l)HX'\W2OlAH)F(F4F(F-*&"fr!)[FhL`$)=J)pz'GpSiMRpR/-`!ebFQ"pe@h())*pkY!exepS\lp)o2y]<'>v>GlQ&=s]$F(F8F(F-*&"fr+s%p&e-&f9*H-6aSI(oj'>>d>4aY"G&>xa(y"Qcf!*ypA7@D-PHG2RS)HMA?FFga?.ujF(F<F(F(*&"hrSu$eB<-&oh[(>mrjMC$z*y6Iw0'Q7F!fCs]d>`Faa**f/0&fy[mYNL@ez#e'=]#[415)e"F(F@F(F(*&"hr!))e3i]%45VXB"4(QZ&p2"\`1n8E)H(*e[l">v4dCDW!=HnmBsts$o`J!*4%o25A*QN;`vG*F(FDF(F(*&"ir!G<C>IS=)Hfc8XfvmS$oxiqB@_idgg)[=.Q0H&p%RQg^tf?]sG-?N,l)*['=xEf!ea%4$F(FHF(F(*&"irgL!ztvdYZ#=:DZO"**\m'y*f:(QE%GY"4*=clSOnOI#[6_F;J"e?BnT18GK*G$*z6QV7hElF(FLF(F(*&"irgp&o!G="oA](y+$fr#p7/LN6g,i-'\w/x<)3!=j!fHAIE&="o.T*HsM+(HMmrWd&\,Auzm)F(FPF(F(*&"ir+gxvPc'=:hDg1cj>*3;Ox3)4Qmh88:O+L"pBOTBLLo$4G@dp(*Q%zxZ]?2]&)zv="Hsl'F(F*F(F("ir+Sm,20mGw&H.Dm#z(*=[Uy#zD`Z_gSh<$p(G)zaCaH>L^78dYfYw[0*)QT6+*oQf;2VE#F(F(-FU6#,&F*F(FQF(F(F(*&,:*&"br+kw$>Y6#*Q=C%4jwx$p"o7HGV0w'Gf7!p*RI8kU&)>"yS:kX6U#\I?/(3(H:I#f:y-/#F(F)F(F(*&"cr?z*G3_!*395Y()4nHqd))pJ#=a%yP_$yHQj7h9^O+$phr_a1,u#\gP%4e<y)z&)>SrGcF(F0F(F(*&"dr+cID#ywLVEpX1*R4_v@k*4:9Le8)R&pfQ8y\U`y4Dd2e=^%3wW$)pc`JgV$Rj#[W![F(F4F(F(*&"erSu>>]0![Q]C%fb*=rPD"e5XnG0!G"*zz?oj7oZHcq#eB8[T>eUC-+5ocy>PBM')HR;F(F8F(F-*&"fr!onFBD:--Df_f)44xhdH+dmTa#**p*f&Rf4neuh'Rp"\!*)4AwtZ>[.=w\-?Eejv2H$pF(F<F(F-*&"gr+ohoc;'*3nm'Gw`*G-gnn6\-b&)[Ke8c"yM(*>x,"**3Z+j%RBe>UI&[f\m&QUR]Z&[]YoF(F@F(F-*&"hr+sm_Uh=r$fMLA7"G%yn4L;N!4yLDX![0&>7"*[(*))Ghjx!**Q4MU%4EPO.7x0UlbGmOc%QF(FDF(F-*&"irgLYz[]`y"3Q:1B>X$z7A2J7]Bh-tmq`#RGAu+7mX,?'=p=^/w1L(p.$p")=V7UNpL"R"F(FHF(F-*&"ir?bVk&H$))R$y>H$[l[FM'3/lXw4FAN0[Yo2[wJ)G^>VzC.^#>E*))f#>!R/.(fEx#R[)="=LF(FLF(F-*&"ir?$G9sn(pA!yVJp(***)y%y&Ql6NHz"e=chdeLi&yCZVoxu4^([e?*p11zXsm>Ji]Ga%3)o]F(FPF(F-*&"ir+ov1O\"\6MK!>h'pYT#zK^9``Kh6:')\>9.*Go$\*y_a()3#*H>>1/a:=$)H^!eudw^w%G^%F(F*F(F-"ir+g,S="3#\G`iD0I3LJcb`PE[2i;+A<(4$RfH7&fGO_w%oQiO88@)H'f*\^K@Ev(>8U&y"F-F(-FU6#,&F*F(FMF(F(F(*&,:*&"`rgLe'3+ZxgLVj!*[#[%>LDC$f9T!3t"3x(G&z[3+qZF&>r#yYC7I)oa/g&Qhk)>)fa='F(F)F(F-*&"brglk#3Qy)=$yQe\\'>`l#eHF.Ik%o&[Njnys'o\I?n_>&=;>`VU,ai4=U#pFr!3.(Q"F(F0F(F-*&"brgx2(\k)yM(4i[616R$RS+^%HYo/ha%3!GBq)o)3LER\ti=zhlu<"\xf&zb(z)Q%o^t(F(F4F(F(*&"dr?j]51Unq`*=2DW"R/'[]&oQ\Q]VlrRGy\i#yw[M]V;0e`I+F7VsvV*Ri9Na&G#RR'F(F8F(F(*&"fr+O1g))Go%G8r\q^[p["*G7U2]dvF6\<pm7**Qgw!G4u&)*Q["R<z:?_%4C)\iC1YG%\d5F(F<F(F(*&"fr!)3!))**3k$H(3i3hf4Yg#3vm.MgQ)4U$[uF:U#H0FU_i:Yd)GufEQ^s&R,H`YZ>dC<]'*F(F@F(F(*&"grSU9MO#4MCognV#e)R3il")R"RDnW^E/34t*42qZ&*Q$=CaKm9"\%f$GU9<3t&R9X/JMScF(FDF(F(*&"hr!)oU(QP&f=&p7")=_;>GqD5L,5'R6*G*e0+_4%)pZWU7qF-O0p4k]jlz'Q%y%y;+Yn")*o?#F(FHF(F(*&"hrg4]6QZH(p`;;PtdI\JLq$3X+fsxsT(>&)[\'oOPaLs&o3q)\&f'z7k"pDw1@CT!)o_O"y&F(FLF(F(*&"hr!o**GqBv'RyVl8=[w>.\$>J=rU(Q-Py'H#3a=Pps]:q>_lRAS%>y&4A&\#)G\RxFB4]j(*F(FPF(F(*&"hr+)[Z!=e[u#fw5it^*=A^[x\2V$pp(3+:k"zh%y')["[P"\i%elgs6+Fmp&f-N^&*e_kfxi*F(F*F(F("hr+?RDi"3YJ%oA*QQCl9q\p.3g3-G.6>HC;#HcX4IWqkS2H:0%*>uTHcYK++W1,N!)e@%F(F(-FU6#,&F*F(FIF(F(F(*&,:*&"^r;P#zl6AH`?k%Gga(3IMy!)>$pDW=c1P@y(G!4D9bz9%\Hd=Q!)3Y_/CL6'HgAzl%F(F)F(F(*&"`r5j)*yP?`jxzGwFs+=%)=$3:z*>ow[S#4R*))3W8Pj!)3`7I5#)*e*o["yzE40^9E>B=F(F0F(F-*&"brgeOg;')QU.![w5;i]d#3yA"Gu$*33OMrO-<d97S]W!>mVjCnA+W)p%=)Rc!oW,5(Q;F(F4F(F-*&"crqabVPjz%***G"\DV9Zg/bD"3K=$))HW#ytHy&)y[[P#)z(zRJ9C^eFubEgWAAH:$fWDVF(F8F(F-*&"dr7W["))\Gg(e1IvadDG&p+\)zr$o8'4U#f(*RRe$pn")=/WJ$\*Ge#GMW_VG-ety(pv'**RiF(F<F(F-*&"erIz6_d?s7I,x1l7S]k')ppFAgpNAPlK0`YQdYw+y][X<\`b.n&o/]0#y)Rr63#[.dF(F@F(F-*&"frSdqvW)*\:i(>eE5AR,RWy;VCBNUhw[t^eqQ(G<-5`Zz#)f/^s4k)GdB#=>'o$[**>g\$F(FDF(F-*&"grI()*QNb\t&\D5$*oh[")p3YQPVTO*oigW*R"zH>P:lfsDykFF$))G+\+$oW/O"=b5s5Ei9F(FHF(F-*&"gr/HUX5]f:JAE69H3Y*z!z`&ohM"G5T=9Y:"*H7s0HgZHn!\0miSKt*3+80eAAET!4&GTF(FLF(F-*&"grg^6f*\m4*4zaY#Hq`'G!yji.jvQ[!*)=tdi,RSS\?B&*o5+eMHx)*zwmh$*eEMvO?\`MvF(FPF(F-*&"gr+%e[97a1j=WCS6BJ5].5dUCn=#>!z#H+N$obGnJZ(RVH]*ftEQ]PSHq^vvk\unKa-)F(F*F(F-"gr+gxb"\q*Ga'>T^S[+uh-A8:^7"e%*GXN]9O%)\xA65/YV1_&\q8%pME6u(zX'z^)z(e'y$F-F(-FU6#,&F*F(FEF(F(F(*&,:*&"\raYB_3!fw*z@Q1u$fWTa)=Ye#e]4(ywir/??(pr)*\%H*o>a0Oi-_y/MTR+v:i))F(F)F(F(*&"^rNDP`]=6O>\P*4/z$)*y4V!e,k\ylT!eP*omS*RY:*ez_'))*R>#ei.a)Go$QVm\w&**F(F0F(F(*&"`rXkYMA%[j$o#[HV'H>K$[f&)z#y^nNvMDjYN>2&3(\Z-i/%QAG<C3)[jPUN.&)=&3NTF(F4F(F(*&"arD%Q%HfA^aalB$y5P%*[fa_,=;NO`5#GW7hHo<O^X"o#zH4Ed"zE5S0#\Id2FqV)G*F(F8F(F(*&"crK*H)))zL$3(\kp;!Q)*>bo#f(HOdUxwu&RZM\t)yj*et=:gWxmHa:)z(Gr$yh]SlMs=I"F(F<F(F(*&"dr&)[031MmX1s/KsV6L\>mK^I"[]kinj,#>"p4Km]/iAp)p0%>BoGD#H..i!e(4I[l@"F(F@F(F(*&"dr0*G8eW-HR\1\kBfm#fU)yb_(*p(zL%Q_FO>q(H=3/JEU'3MTmO0FHk9ZZl+\.mG0yF(FDF(F(*&"ervFnH5t"\,B/Dq'[wD'**\!pm"f%e0277QL`r)4zv9lj!)yv7ZAKb%H$p(eNfdla4i)fX$F(FHF(F(*&"frkcjr0<w)=R?;+DfTAf@K0Xr-H>CJ.KNn"G***eFZUtl=`6X`:jfC`#\F">Qw(z3z."F(FLF(F(*&"fr!)p7*Hf(*>Uqw-)3G*\CeG^7G><wj#RVnjMmR5o+-$))Ht-'RF'oFs<9v8zT&)e@2Jq;?F(FPF(F(*&"fr+oO-W#***=r)e^k()\J0?,(yn,\U^!*f?HO--G$[x4#3=363b&**>>)el#)oq"=h\F#z4MG#F(F*F(F("fr++[m5QktO23Z**p'zIkGu'oLa!**p*z1gRQqdus'pW$p@.7)Gl,RHl$yg&>#y_^'RhWT6F(F(-FU6#,&F*F(FAF(F(F(/-FU6#""!F\y/-FU6#F(F\y/-FU6#FQFM/-FU6#FM"#X/-FU6#FI"$%f/-FU6#FE"%Xx

rec1:=map(factor,op(1,rec)):

(hypergeomsols(rec1,co(k),{co(0) = 0, co(1) = 0, co(2) = 3, co(3) = 45, co(4) = 594, co(5) = 7745}));

hypergeomsols(rec1,co(k),{});

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

hypergeomsols(rec1,co(k),{co(0) = 594, co(1) = 7745});

LCYqOCMiJ0VnPCIjIikiIiIpSSNQaUclKnByb3RlY3RlZEcjRiciIiMhIiIpIiIkRitGJy1JJkdBTU1BRzYkRipJKF9zeXNsaWJHNiI2IyNGLEYvRi0tRjE2IyMiIiYiIidGJykiIztJImtHRjRGJy1GMTYjLCZGPkYnRitGJ0YnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RicjRkZGL0YnRi0tRjE2IywmRj5GJ0YvRidGLSwoKiQpRj5GLEYnRicqJiIiJUYnRj5GJ0YnIyIjNkYvRidGJ0YnKjYjIiYuWSciI2FGJ0Y3Ri1GKEYtRjBGJ0Y8RictRjE2IywmRj5GJ0Y5RidGJ0Y/RictRjE2IywmRj5GJyMiIilGL0YnRi1GS0YtLChGT0YnKiZGO0YnRj5GJ0YnIyIjbCIiKkYnRidGLQ==

seq(%,k=4..10);

NikjIilsSnNoIiIpIikqcCs6KiIrKG9AXDgiIyIsQyQ0QnhzIiImIyIuLFRBJkdJUSIjPyIubWYySD9kIyMiMDg7OChSQT5jIiM7