<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*(x+1/x)*Cm(1/x)-t/x*(y+1/y)*Cm(1/y)-t/x/y*C0;

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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:

C0ser:=proc(n) local res: res:=coeff(convert(Cser(n),polynom),y,0): coeff(coeff(res,x,0),y,0): end:

Cser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C0=C0ser(n-1),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));

KytJInRHNiIiIiIiIiEsKComSSJ4R0YkRiVJInlHRiRGJUYlKiZGKUYlRiohIiJGJSomRilGLEYqRiVGJUYlLDIqJilGKSIiI0YlKUYqRjFGJUYlKiZGMUYlRjBGJUYlKiZGMUYlRjJGJUYlKiZGMEYlRjJGLEYlIiIkRiUqJkYwRixGMkYlRiUqJEYyRixGJSokRjBGLEYlRjEsOiomKUYpRjZGJSlGKkY2RiVGJSooRjZGJUYqRiVGPEYlRiUqKEY2RiVGKUYlRj1GJUYlKihGNkYlRjxGJUYqRixGJSooIiIpRiVGKUYlRipGJUYlKihGNkYlRilGLEY9RiVGJSomRjxGJUY9RixGJSooIiIoRiVGKUYlRipGLEYlKihGRkYlRilGLEYqRiVGJSomRjxGLEY9RiVGJSooRjFGJUYpRiVGPUYsRiUqKEYxRiVGPEYsRipGJUYlRjY=

All walks in C: this sequence does not appear in the OEIS

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 either

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*(x+1/x)*Qp(x)-t/x*(y+1/y)*Qp(y)+t/x/y*Q0;

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Expansions

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

Q0ser:=proc(n) subs(x=0,y=0,Qser(n)): end:

Qser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q0=Q0ser(n-1),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);

Ky1JInRHNiIiIiIiIiEqJkkieEdGJEYlSSJ5R0YkRiVGJSomLCYqJClGKCIiI0YlRiVGJUYlRiUsJiokKUYpRi5GJUYlRiVGJUYlRi4qKEYoRiVGKUYlLCoqJkYtRiVGMUYlRiUqJkYuRiVGLUYlRiUqJkYuRiVGMUYlRiUiIiVGJUYlIiIkKihGK0YlRi9GJUYzRiVGNw==

<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) (i+1)*(j+1)/(1+(i+n)/2)/(1+(j+n)/2)*(n)!^2/((n-i)/2)!/((n+i)/2)!/((n-j)/2)!/((n+j)/2)!: end:

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

NikiIiJGIyIiJSIjRCIkJz4iJWs8IiZDdSI=

i:=2: j:=4: n:=28: coeff(coeff(coeff(convert(Qser(n),polynom),x,i),y,j),t,n)-Qconj(i,j,n);

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unassign('n'):Qconj('i','j','n');

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Qconj(0,0,2*k);

KigpLCYiIiJGJUkia0c2IkYlIiIjISIiKS1JKmZhY3RvcmlhbEclKnByb3RlY3RlZEc2IywkKiZGKEYlRiZGJUYlRihGJSktRiw2I0YmIiIlRik=

<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);

LCwqKEkieEc2IiIiIkkieUdGJUYmLUkiUUdGJTYkRiRGJ0YmRiYqKEYnRiZGJCEiIi1GKTYkKiRGJEYsRidGJkYsKihGJEYsRidGLC1GKTYkRi8qJEYnRixGJkYmKihGJEYmRidGLC1GKTYkRiRGM0YmRiwqJiwmRiRGJkYvRixGJiwmRidGJkYzRixGJkYs

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);

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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);

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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

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));

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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);

LCYtSSNBbUc2IjYjKiRJInhHRiUhIiIiIiIqKEkiYUdGJUYqKUYoIiIjRiktSSNRcEdGJUYmRipGKQ==

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

LCYtSSNBbUc2IjYjKiRJInlHRiUhIiIiIiIqKEkiYUdGJUYqLUkjUXBHRiVGJkYqKUYoIiIjRilGKQ==

eqC;

eqA:=collect(subs(C0=A0+(1-2*a)*Q0,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,Q0,A0],factor);

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To get rid of Qp, we take a=1/3

eqA:=collect(subs(a=1/3,eqA),[A,Am,A0],factor);

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expand(op(5,eqA));

LCgjIiIjIiIkIiIiKiYjRiZGJUYmKUkieUc2IkYkISIiRiYqJkYoRiYpSSJ4R0YrRiRGLEYm

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:

A0ser:=proc(n) coeff(coeff(convert(Aser(n),polynom),y,0),x,0):end:

n:=16: normal(series(subs(A0=A0ser(n),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:

eqA := (t*x^2*y^2+t*x^2+t*y^2-x*y+t)*A(x, y)/(x*y)-t*(x^2+1)*Am(1/x)/(x*y)-t*(y^2+1)*Am(1/y)/(x*y)-t*A0/(x*y)+(1/3)*(2*x^2*y^2+x^2+y^2)/(x^2*y^2);

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKC1JIkFHRic2JEYqRi1GKEYqRjFGLUYxRigqLEYmRigsJiokRilGKEYoRihGKEYoLUkjQW1HRic2IyokRipGMUYoRipGMUYtRjFGMSosRiZGKCwmKiRGLEYoRihGKEYoRigtRjk2IyokRi1GMUYoRipGMUYtRjFGMSoqRiZGKEkjQTBHRidGKEYqRjFGLUYxRjEqKiNGKCIiJEYoLCgqKEYrRihGKUYoRixGKEYoRjdGKEY+RihGKEYpRjFGLEYxRig=

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);

LCwqLCMiIiIiIiRGJUkieEc2IkYlLCYqJClJInlHRigiIiNGJUYlRiVGJUYlLC4qJilGJyIiJUYlRitGJUYlKiZGLUYlRjBGJUYlKigiIidGJSlGJ0YtRiVGK0YlRiUqJiIjN0YlRjVGJUYlKiYiIzlGJUYrRiVGJSIjSkYlRiUpSSJ0R0YoRjFGJUYlKipGJEYlRixGJSwuRi9GJUYyRiUqKCIiJkYlRjVGJUYrRiVGJSomIiM1RiVGNUYlRiUqJiIiKkYlRitGJUYlIiNARiVGJSlGPEYmRiVGJSomLCoqKCNGMUYmRiVGJ0YlRitGJUYlKiZGSkYlRidGJUYlKihGJEYlKUYnRiZGJUYrRiVGJSomRiRGJUZNRiVGJUYlKUY8Ri1GJUYlKiYsJkYsRiUqKEYkRiVGNUYlRixGJUYlRiVGPEYlRiUqJkYkRiVGJ0YlRiU=

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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<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);

LCoqKEkieEc2IiIiIkkieUdGJUYmLUkiQUdGJTYkRiRGJ0YmRiYqKEYnRiZGJCEiIi1GKTYkKiRGJEYsRidGJkYsKihGJEYsRidGLC1GKTYkRi8qJEYnRixGJkYmKihGJEYmRidGLC1GKTYkRiRGM0YmRiw=

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));

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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)));

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

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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));

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Let us check with the series

n:=3:Pser(n);

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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);

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

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Expression of A0

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

LCQqJiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiU2I0kiTUdGLTYkIiIhRjJGJUYl

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),A0=2*M10, 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);

PC5JJE0xMEc2IkkidEdGJEkieEdGJEkieUdGJC1JIk1HRiQ2JCIiIUYmLUYpNiRGK0YnLUYpNiRGJkYnLUYpNiRGJ0YmLUYpNiQqJEYmISIiRistRik2JEY0RictRik2JCokRidGNUYrLUYpNiRGOkYm

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)), M10=coeff(coeff(Mser(n),x,1),y,0), 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; recall that M(0,0)=0

subs(x=0,y=0,Mser(n));

IiIh

eqM;

LDYqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKEYqRjFGLEYxLUkiTUdGJzYkIiIhRipGKEYxKipGJEYoRilGMUYtRjEtRjM2JEY1Ri1GKEYxKipGJEYoRilGMUYtRjEtRjM2JEYqRi1GKEYoKipGJEYoRipGMUYsRjEtRjM2JEYtRipGKEYoKixGJkYoLCYqJEYpRihGKEYoRihGKEYpRjFGLUYxLUYzNiQqJEYqRjFGNUYoRjEqKkYkRihGKUYxRi1GMS1GMzYkRkRGLUYoRigqKkYkRihGKkYxRixGMS1GMzYkKiRGLUYxRipGKEYoKixGJkYoLCYqJEYsRihGKEYoRihGKEYqRjFGLEYxLUYzNiRGS0Y1RihGMSosRitGKEYmRihGKkYxRi1GMUkkTTEwR0YnRihGMSoqI0YoIiIkRigsKCooRitGKEYpRihGLEYoRihGQUYoRk5GKEYoRilGMUYsRjFGKA==

Some series in x contribute for a few terms

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

KylJInhHNiIjIiIiIiIkISIjLCQqKEkidEdGJEYmLCwqJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiY2I0kiTUdGJDYkIiIhSSJ5R0YkRiYpRjpGJ0YmISIiKiYtRjc2JEY6RjlGJilGOiIiI0YmRjwqKEZBRiZJJE0xMEdGJEYmRjpGJkYmKiZGLkYmRjpGJkY8Rj5GPEYmRkBGPEY8RjwtSSJPR0YzRjVGOQ==

and we must remove from the series in 1/x a few non-negative terms

op(5,eqM)+op(6,eqM);

LCYqLEkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJkYpISIiSSJ5R0YlRiwtSSJNR0YlNiQqJEYqRiwiIiFGJkYsKiosLCooRiRGJkYpRiYpRi1GK0YmRiYqJkYkRiZGKUYmRiYqJkYkRiZGNkYmRiYqJkYqRiZGLUYmRixGJEYmRiZGKUYsRi1GLC1GLzYkRjFGLUYmRiY=

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

KytJInhHNiIqKkkidEdGJCIiIiwmKiQpSSJ5R0YkIiIjRidGJ0YnRidGJy1JIk1HRiQ2JCIiIUYrRidGKyEiIkYwLCQqJiwqKigtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnNiNGLkYvRidGJkYnRipGJ0YxKiZGJkYnLUY3NiRGMEYwRidGJyomRiZGJ0Y2RidGMSomRitGJ0YtRidGJ0YnRitGMUYxRicsJCooI0YnRixGJywuKigtLSZGOTYkRidGJ0Y+Ri9GJ0YmRidGKkYnRjEqKkYsRidGLUYnRiZGJ0YqRidGMSomRiZGJy1GSkZBRidGJyooRixGJ0Y2RidGK0YnRicqJkYmRidGSUYnRjEqKEYsRidGJkYnRi1GJ0YxRidGK0YxRjFGLC1JIk9HRjtGPSIiJA==

eqMxneg:=convert(%%%,polynom)+op(5,eqM)+op(6,eqM)- coeff(%,x,0);

LCwqJiMiIiIiIiRGJSlJInhHNiIiIiMhIiJGJSoqSSJ0R0YpRiUsLComLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGJTYjSSJNR0YpNiQiIiFJInlHRilGJSlGPEYmRiVGKyomLUY5NiRGPEY7RiUpRjxGKkYlRisqKEYqRiVJJE0xMEdGKUYlRjxGJUYlKiZGMEYlRjxGJUYrRj9GK0YlRkFGK0YoRitGKyosRi1GJSwmKiRGJ0YlRiVGJUYlRiVGJ0YrRjxGKy1GOTYkKiRGKEYrRjtGJUYrKiosLCooRi1GJUYnRiVGQUYlRiUqJkYtRiVGJ0YlRiUqJkYtRiVGQUYlRiUqJkYoRiVGPEYlRitGLUYlRiVGJ0YrRjxGKy1GOTYkRkpGPEYlRiUqKkYtRiUsJiokRkFGJUYlRiVGJUYlLUY5RjpGJUY8RitGKw==

indets(eqMxneg);

PCtJJE0xMEc2IkkidEdGJEkieEdGJEkieUdGJC1JIk1HRiQ2JCIiIUYnLUYpNiRGJ0YrLUYpNiQqJEYmISIiRistRik2JEYwRictLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIjYjRilGKg==

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

normal(subs(M(0,0)=0,series(normal(eqM-eqMxneg),x,3)));

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

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

normal(subs((D[1](M))(0, 0)=M10,M(0,0)=0, series(subs(x=1/x,eqMxneg),x,3)));

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

The coefficient of 1/x in eqMxneg

com1:=coeff(%,x):

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

LCQqLCMiIiIiIiNGJSwqKigtSSJNRzYiNiRJInlHRisiIiFGJUkidEdGK0YlKUYtRiZGJSEiIioqIiIkRiVGL0YlSSRNMTBHRitGJUYtRiVGJSomLUYqNiRGLkYtRiVGMEYlRiUqJkYpRiVGL0YlRjFGJUYtRjFGL0YxLCYqJEYwRiVGJUYlRiVGMUYl

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

LCwqLCMiIiIiIiNGJSwoKipGJkYlSSJ0RzYiRiVJInhHRipGJSlJInlHRipGJkYlRiUqKEYmRiVGKUYlRitGJUYlRi0hIiJGJUYtRi9GK0YvLUkiTUdGKjYkIiIhRi1GJUYvKi5GJEYlRilGJSwmKiRGLEYlRiVGJUYlRiVGK0YvRixGLy1GMTYkRi1GM0YlRiUqLEYpRiUsJiokKUYrRiZGJUYlRiVGJUYlRjxGL0YtRi8tRjE2JCokRitGL0YzRiVGLyoqLCwqKEYpRiVGPEYlRixGJUYlKiZGKUYlRjxGJUYlKiZGKUYlRixGJUYlKiZGK0YlRi1GJUYvRilGJUYlRjxGL0YtRi8tRjE2JEY/Ri1GJUYlKiojRiUiIidGJSwmKioiIiRGJUkkTTEwR0YqRiVGKUYlRitGJUYlKiZGJkYlRi1GJUYvRiVGLUYvRjxGL0Yv

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

LCwqKiMiIiIiIiNGJSwoKihGJkYlSSJ0RzYiRiUpSSJ5R0YqRiZGJUYlKiZJInhHRipGJUYsRiUhIiIqJkYmRiVGKUYlRiVGJS1JIk1HRio2JCIiIUYsRiVGLEYvRi8qLkYkRiVGKUYlLCYqJEYrRiVGJUYlRiVGJUYuRiVGK0YvLUYyNiRGLEY0RiVGJSoqRilGJSwmKiQpRi5GJkYlRiVGJUYlRiUtRjI2JEYuRjRGJUYsRi9GLyooLCwqKEYpRiVGPUYlRitGJUYlKiZGKUYlRj1GJUYlKiZGKUYlRitGJUYlRi1GL0YpRiVGJS1GMjYkRi5GLEYlRixGL0YlKiojRiUiIidGJSwmKigiIiRGJUkkTTEwR0YqRiVGKUYlRiUqKEYmRiVGLkYlRixGJUYvRiVGLkYlRixGL0Yv

Let us check

indets(eqcatM);

PCpJJE0xMEc2IkkidEdGJEkieEdGJEkieUdGJC1JIk1HRiQ2JCIiIUYnLUYpNiRGJkYrLUYpNiRGJkYnLUYpNiRGJ0Yr

n:=7: normal(series(subs( M10=coeff(subs(y=0,Mser(n)),x,1), 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));

LCQqKiMiIiIiIidGJSw0KixGJkYlLUkiTUc2IjYkSSJ4R0YrIiIhRiVJInRHRitGJSlGLSIiI0YlSSJ5R0YrRiVGJSosIiIkRiUtRio2JEYyRi5GJUYvRiVGLUYlKUYyRjFGJSEiIioqRiZGJS1GKjYkRi5GMkYlRi9GJSlGMkY0RiVGJSosRjRGJUYvRiVJJE0xMEdGK0YlRi1GJUYyRiVGJSoqRjRGJUY6RiVGLUYlRjdGJUY4KihGMUYlRjBGJUY3RiVGOCoqRiZGJUYvRiVGKUYlRjJGJUYlKipGNEYlRjVGJUYvRiVGLUYlRjgqKkYmRiVGOkYlRi9GJUYyRiVGJUYlRjJGOCwsKihGL0YlRjBGJUY3RiVGJSomRi9GJUYwRiVGJSomRi9GJUY3RiVGJSomRi1GJUYyRiVGOEYvRiVGOEYl

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

eqcatM;

a slightly different way of writing this equation: we introduce N=2M(x,y)-M(0,y)

factor(subs(t=0,eqcatM));

LCQqKCMiIiIiIidGJUkieEc2IkYlLCgqJiIiJEYlLUkiTUdGKDYkIiIhSSJ5R0YoRiUhIiIqJkYmRiUtRi02JEYnRjBGJUYlKiYiIiNGJUYnRiVGMUYlRjE=

unassign('N'):

eqcatN:=collect(subs(M(x,y)=1/2*(N(x,y)+M(0,y)),2*eqcatM/x),[N,M,M10],factor);

LC4qKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKEYqRjFGLUYxLUkiTkdGJzYkRipGLUYoRigqMEYmRigsJiokRixGKEYoRihGKEYoLCZGKkYoRihGMUYoLCZGKkYoRihGKEYoRipGMUYtRjEtSSJNR0YnNiQiIiFGLUYoRigqLkYrRihGJkYoLCYqJEYpRihGKEYoRihGKEYqRjFGLUYxLUY7NiRGKkY9RihGMSoqRiZGKEY2RihGLEYxLUY7NiRGLUY9RihGKCooRiZGKEYtRjFJJE0xMEdGJ0YoRjEqJiNGKyIiJEYoRipGKEYo

eqpap:=K*(2*M(x,y)-M(0,y))-2*x/3+2*t/y*(x+1/x)*M(x,0) -t*(x-1/x)*(y+1/y)*M(0,y)-t*(1+1/y^2)*M(y,0)+t/y*M10;

LC4qJiwmIiIiRiUqKEkidEc2IkYlLCZJInhHRihGJSokRiohIiJGJUYlLCZJInlHRihGJSokRi5GLEYlRiVGLEYlLCYqJiIiI0YlLUkiTUdGKDYkRipGLkYlRiUtRjQ2JCIiIUYuRixGJUYlKiYjRjIiIiRGJUYqRiVGLCosRjJGJUYnRiVGLkYsRilGJS1GNDYkRipGOEYlRiUqKkYnRiUsJkYqRiVGK0YsRiVGLUYlRjZGJUYsKihGJ0YlLCYqJClGLkYyRixGJUYlRiVGJS1GNDYkRi5GOEYlRiwqKEYnRiVGLkYsSSRNMTBHRihGJUYl

factor(eqcatM/eqpap);

LCQqJiMiIiIiIiNGJUkieEc2IkYlISIi

Results of the above sub-section

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

LCoqJkkieEc2IiEiIi1JIk1HRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1GKDYkRipGJEYrRisqJkYqRiYtRig2JEYvRiRGK0Ym

eqcatM:=K*(2*M(x,y)-M(0,y))-2*x/3+2*t/y*(x+1/x)*M(x,0) -t*(x-1/x)*(y+1/y)*M(0,y)-t*(1+1/y^2)*M(y,0)+t/y*M10;;

LC4qJiwmIiIiRiUqKEkidEc2IkYlLCZJInhHRihGJSokRiohIiJGJUYlLCZJInlHRihGJSokRi5GLEYlRiVGLEYlLCYqJiIiI0YlLUkiTUdGKDYkRipGLkYlRiUtRjQ2JCIiIUYuRixGJUYlKiYjRjIiIiRGJUYqRiVGLCosRjJGJUYnRiVGLkYsRilGJS1GNDYkRipGOEYlRiUqKkYnRiUsJkYqRiVGK0YsRiVGLUYlRjZGJUYsKihGJ0YlLCZGJUYlKiQpRi5GMkYsRiVGJS1GNDYkRi5GOEYlRiwqKEYnRiVGLkYsSSRNMTBHRihGJUYl

eqcatN:=(t*x^2*y^2+t*x^2+t*y^2-x*y+t)*N(x, y)/(y*x)+t*(y^2+1)*(x-1)*(x+1)*M(0, y)/(x*y)-2*t*(x^2+1)*M(x, 0)/(y*x)+t*(y^2+1)*M(y, 0)/y^2-t*M10/y+(2/3)*x ;

LC4qKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKC1JIk5HRic2JEYqRi1GKEYqRjFGLUYxRigqMEYmRigsJiokRixGKEYoRihGKEYoLCZGKkYoRihGMUYoLCZGKkYoRihGKEYoLUkiTUdGJzYkIiIhRi1GKEYqRjFGLUYxRigqLkYrRihGJkYoLCYqJEYpRihGKEYoRihGKC1GOzYkRipGPUYoRipGMUYtRjFGMSoqRiZGKEY2RigtRjs2JEYtRj1GKEYsRjFGKCooRiZGKEYtRjFJJE0xMEdGJ0YoRjEqJiNGKyIiJEYoRipGKEYo

The roots of the kernel

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

NyQsJCoqIyIiIiIiI0YmLCZJInhHNiJGJiokKSwkKiYsKCooRidGJkkidEdGKkYmKUYpRidGJkYmKiZGJ0YmRjFGJkYmRilGJkYmLChGMEYmRjNGJkYpISIiRiZGNUYlRiZGJkYmRjFGNSwmKiRGMkYmRiZGJkYmRjVGJiwkKipGJUYmLCZGKUY1RitGJkYmRjFGNUY2RjVGNQ==

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

LCQqKiMiIiIiIiNGJSwmSSJ4RzYiISIiKiQpLCQqJiwoKihGJkYlSSJ0R0YpRiUpRihGJkYlRiUqJkYmRiVGMUYlRiVGKEYlRiUsKEYwRiVGM0YlRihGKkYlRipGJEYlRiVGJUYxRiosJiokRjJGJUYlRiVGJUYqRio=

LCQqKiMiIiIiIiNGJSwmSSJ4RzYiRiUqJCksJComLCgqKEYmRiVJInRHRilGJSlGKEYmRiVGJSomRiZGJUYwRiVGJUYoRiVGJSwoRi9GJUYyRiVGKCEiIkYlRjRGJEYlRiVGJUYwRjQsJiokRjFGJUYlRiVGJUY0RiU=

The discriminant is an even function of t and x

Delta:=(1-2*t*(x+1/x))*(1+2*t*(x+1/x));expand(Delta);collect(%,t,factor);

KiYsJiIiIkYkKigiIiNGJEkidEc2IkYkLCZJInhHRihGJCokRiohIiJGJEYkRixGJCwmRiRGJEYlRiRGJA==

LCoiIiJGIyooIiIlRiMpSSJ0RzYiIiIjRiMpSSJ4R0YoRilGIyEiIiomIiIpRiNGJkYjRiwqKEYlRiNGJkYjRipGLEYs

LCYiIiJGIyoqIiIlRiMpLCYqJClJInhHNiIiIiNGI0YjRiNGI0YsRiNGKSEiIilJInRHRitGLEYjRi0=

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

LCQqKiMiIiIiIiNGJSwmRiVGJSokKSomLCZGJUYlKihGJkYlSSJ0RzYiRiUsJkkieEdGLkYlKiRGMCEiIkYlRiVGMkYlLCZGJUYlRixGJUYlRiRGJUYyRiVGLUYyRi9GMkYl

simplify(Ypap/Y0sol,symbolic);

IiIi

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

Y0ser(4);

KylJInRHNiIqJiwmKiQpSSJ4R0YkIiIjIiIiRitGK0YrRitGKSEiIkYrKiYpRikiIiRGLClGJkYvRitGLy1JIk9HJSpwcm90ZWN0ZWRHNiNGKyIiJg==

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>

Use (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);

LCwqMEkidEc2IiIiIiwmKiQpSSJZR0YlIiIjRiZGJkYmRiZGJiwmSSJ4R0YlRiZGJiEiIkYmLCZGLUYmRiZGJkYmRi1GLkYqRi4tSSJNR0YlNiQiIiFGKkYmRi4qLkYrRiZGJEYmLCYqJClGLUYrRiZGJkYmRiZGJkYtRi5GKkYuLUYxNiRGLUYzRiZGJioqRiRGJkYnRiZGKUYuLUYxNiRGKkYzRiZGLiooRiRGJkYqRi5JJE0xMEdGJUYmRiYqJiNGKyIiJEYmRi1GJkYu

LCwqMEkidEc2IiIiIiwmKiQpSSJZR0YlIiIjRiZGJkYmRiZGJiwmKiRJInhHRiUhIiJGJkYmRi9GJiwmRi1GJkYmRiZGJkYuRiZGKkYvLUkiTUdGJTYkIiIhRipGJkYvKi5GK0YmRiRGJiwmKiQpRi5GK0YvRiZGJkYmRiZGLkYmRipGLy1GMjYkRi1GNEYmRiYqKkYkRiZGJ0YmRilGLy1GMjYkRipGNEYmRi8qKEYkRiZGKkYvSSRNMTBHRiVGJkYmKiYjRisiIiRGJkYuRi9GLw==

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

LCwqMEkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJiwmSSJZR0YlRiZGJiEiIkYmLCZGLUYmRiZGJkYmRi1GLkYqRi4tSSJNR0YlNiQiIiFGKkYmRi4qLkYrRiZGJEYmLCYqJClGLUYrRiZGJkYmRiZGJkYtRi5GKkYuLUYxNiRGLUYzRiZGJioqRiRGJkYnRiZGKUYuLUYxNiRGKkYzRiZGLiooRiRGJkkkTTEwR0YlRiZGKkYuRiYqJiNGKyIiJEYmRi1GJkYu

LCwqMEkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjISIiRiZGJkYmRiYsJkkiWUdGJUYmRiZGLEYmLCZGLkYmRiZGJkYmRi5GLEYqRiYtSSJNR0YlNiQiIiEqJEYqRixGJkYsKi5GK0YmRiRGJiwmKiQpRi5GK0YmRiZGJkYmRiZGLkYsRipGJi1GMTYkRi5GM0YmRiYqKkYkRiZGJ0YmRilGJi1GMTYkRjRGM0YmRiwqKEYkRiZJJE0xMEdGJUYmRipGJkYmKiYjRisiIiRGJkYuRiZGLA==

indets(f1*f2*f3*f4);

PCxJJE0xMEc2IkkiWUdGJEkidEdGJEkieEdGJC1JIk1HRiQ2JCIiIUYlLUYpNiRGK0YnLUYpNiRGKyokRichIiItRik2JEYlRistRik2JEYnRistRik2JEYwRis=

We eliminate the series involving Y

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

PCQvLUkiTUc2IjYkIiIhSSJZR0YmLCQqLiMiIiIiIiRGLSwuKipGLkYtLUYlNiRJInhHRiZGKEYtSSJ0R0YmRi0pRjMiIiNGLUYtKipGLkYtLUYlNiQqJEYzISIiRihGLUY0Ri1GNUYtRjsqJkY1Ri1GKUYtRjsqKEYuRi1GMUYtRjRGLUYtKihGLkYtRjhGLUY0Ri1GO0YpRi1GLUY0RjssJkYzRi1GLUY7RjssJkYzRi1GLUYtRjssJiokKUYpRjZGLUYtRi1GLUY7Ri0vLUYlNiRGKUYoLCQqLkYsRi0sMEYwRi1GN0YtKipGLkYtRjRGLUkkTTEwR0YmRi1GM0YtRi1GPEY7Rj1GLUY+Ri1GKUY7Ri1GKUYtRjNGO0Y0RjtGQUY7Ri0=

Here are our two equations

f3bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f3))),[M,M10],factor);

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYsRi0tSSJNR0YoNiQiIiFGJ0YqRi0qKEYkRipGJ0YtLUYwNiRGJ0YyRipGKioqRilGKkYkRipGJ0YtLUYwNiQqJEYnRi1GMkYqRioqJiIiJEYqSSRNMTBHRihGKkYqKiwjRilGO0YqRixGKiwmKiZGKUYqRiZGKkYqRipGKkYqSSJ0R0YoRi1GJ0YtRi0=

f4bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f4))),[M,M10],factor);

LCwqLiwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYnRi1GLEYtLUkiTUdGKDYkIiIhKiRGJ0YtRipGLSomLCYqJkYpRipGJkYqRipGKUYqRiotRjA2JEYnRjJGKkYqKiZGJEYqLUYwNiRGM0YyRipGKiooIiIkRipGJ0YqSSRNMTBHRihGKkYqKiojRilGPUYqRixGKiwmRiVGKkYpRipGKkkidEdGKEYtRi0=

factor(subs(x=1/x,f3bis)/f4bis);

KiRJInhHNiIhIiI=

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));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Eliminate M(1/x,0)

f8:=collect(-f3bis+2/x*f4bis,[M,M10],factor);

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYsRi0tSSJNR0YoNiQiIiFGJ0YqRioqMEYpRipGJkYtRiRGKkYrRipGLkYqRixGLS1GMDYkRjIqJEYnRi1GKkYtKioiIiRGKkYkRipGJ0YtLUYwNiRGJ0YyRipGKiomRjhGKkkkTTEwR0YoRipGKioqRilGKkYsRipGJ0YtSSJ0R0YoRi1GLQ==

What we should remember from the above subsection

f8 := (x^2+1)*(Y-1)*(Y+1)*M(0, x)/Y-(2*(x^2+1))*(Y-1)*(Y+1)*M(0, 1/x)/(x^2*Y)+(3*(x^2+1))*M(x, 0)/x+3*M10-2*Y/(t*x);

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKi1JIk1HRig2JCIiIUYnRipGLEYtRioqMEYpRipGJEYqRitGKkYuRiotRjA2JEYyKiRGJ0YtRipGJkYtRixGLUYtKioiIiRGKkYkRiotRjA2JEYnRjJGKkYnRi1GKiomRjhGKkkkTTEwR0YoRipGKioqRilGKkYsRipJInRHRihGLUYnRi1GLQ==

factor(expand(rationalize(subs(Y=Y0sol,(x^2+1)*(Y-1/Y)))));

LCQqJkkidEc2IiEiIiksJComLCgqKCIiIyIiIkYkRi0pSSJ4R0YlRixGLUYtKiZGLEYtRiRGLUYtRi9GLUYtLChGK0YtRjBGLUYvRiZGLUYmI0YtRixGLUYm

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

LCQqJkkieEc2IiIiIkkidEdGJSEiIkYo

Introducing De=Delta

f8form:=1/t*sqrt(De)*(x*M(0,x)-2/x*M(0,1/x))-3*(x+1/x)*M(x, 0)+2*Y/(x*t)-3*M10;

LCoqKEkidEc2IiEiIilJI0RlR0YlIyIiIiIiI0YqLCYqJkkieEdGJUYqLUkiTUdGJTYkIiIhRi5GKkYqKihGK0YqRi5GJi1GMDYkRjIqJEYuRiZGKkYmRipGKiooIiIkRiosJkYuRipGNkYqRiotRjA2JEYuRjJGKkYmKipGK0YqSSJZR0YlRipGJEYmRi5GJkYqKiZGOEYqSSRNMTBHRiVGKkYm

n:=10: normal(series(subs(De=Delta, M10=coeff(coeff(Mser(n),x,1),y,0),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));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

<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>

M(x,0) is a multiple of x, and M(x,0)/x is even in x (and t, but we won't use this)

normal(series(subs(y=0,1/x*Mser(4)),t));

KylJInRHNiIjIiIiIiIkIiIhLCYjIiIlRidGJiomRiVGJilJInhHRiQiIiNGJkYmRi8sKComI0YvRidGJilGLkYrRiZGJiomRitGJkYtRiZGJiMiI0pGJ0YmRis=

Introduce the series R and S which are even in t and x

Rser:=proc(n) collect(t^2/x*subs(y=0,Mser(n-1)),t,factor): end:

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

Rser(4);Sser(4);

LCYqJiwmIyIiJSIiJCIiIiomI0YoRidGKClJInhHNiIiIiNGKEYoRigpSSJ0R0YtRiZGKEYoKiZGKkYoKUYwRi5GKEYo

LCYqKClJInhHNiIiIiMiIiIsJiomIiIkRihGJEYoRigiIihGKEYoKUkidEdGJiIiJUYoRigqJilGLkYnRihGJEYoRig=

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^2*x,M10=R0/t^2,Y=(1-sqrt(De))/2/t/(x+1/x),t^2*f8form),[De,R],expand);

LCoqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjAqJiwmRilGKkYvRipGMEYpRjBGMEYqKUkjRGVHRicjRipGLEYqRioqJiwmKiYiIiRGKilGKUYsRipGMEY5RjBGKi1JIlJHRidGKEYqRipGMUYqKiZGOUYqSSNSMEdGJ0YqRjA=

n:=6: normal(series(subs(S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)),De=Delta,R(x)=Rser(n),R0=coeff(Rser(n),x,0),f13),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

use parity: S(x,t)=Ss(x^2,t), same for R. We keep t instead of sqrt(t), even though there is parity in t as well.

The series Rs and Ss are those denoted R and S in the paper

Rsser:=proc(n) subs(x=sqrt(x),Rser(2*n)): end:

Rsser(3);

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Ssser:=proc(n) subs(x=sqrt(x),Sser(2*n)): end:

Ssser(3);

LCgqKEkieEc2IiIiIiwoKiYiIzVGJilGJCIiI0YmRiYqJiIjV0YmRiRGJkYmIiNsRiZGJilJInRHRiUiIidGJkYmKihGJEYmLCYqJiIiJEYmRiRGJkYmIiIoRiZGJilGMCIiJUYmRiYqJilGMEYrRiZGJEYmRiY=

Deltas:=collect(subs(x=sqrt(x),expand(Delta)),t,factor);

LCYiIiJGIyoqIiIlRiMpLCZJInhHNiJGI0YjRiMiIiNGI0YoISIiKUkidEdGKUYqRiNGKw==

f13s:=collect(factor((subs(S(x)=Ss(x^2),S(1/x)=Ss(1/x^2),R(x)=Rs(x ^2),x=sqrt(x),De=Des,f13))),[Des,Rs,R0,Ss],factor);

LCoqJiwoLUkjU3NHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwKiQsJkYpRipGKkYqRjBGMEYqKUkkRGVzR0YnI0YqRixGKkYqKiYsJiomIiIkRipGKUYqRjBGOUYwRiotSSNSc0dGJ0YoRipGKiomRjlGKkkjUjBHRidGKkYwRjFGKg==

series(subs(Des=Deltas,f13s),x=-1,1);;

KycsJkkieEc2IiIiIkYmRiYsJi1JI1NzR0YlNiMhIiJGKyomIiIkRiZJI1IwR0YlRiZGKyIiIS1JIk9HJSpwcm90ZWN0ZWRHNiNGJkYm

op(1,f13s);

KiYsKC1JI1NzRzYiNiNJInhHRiYiIiIqJiIiI0YpLUYlNiMqJEYoISIiRilGLyokLCZGKEYpRilGKUYvRi9GKSlJJERlc0dGJiNGKUYrRik=

u0:=op(1,f13s)^2-subs(Des=0,f13s)^2;

LCYqJiksKC1JI1NzRzYiNiNJInhHRigiIiIqJiIiI0YrLUYnNiMqJEYqISIiRitGMSokLCZGKkYrRitGK0YxRjFGLUYrSSREZXNHRihGK0YrKiQpLCgqJiwmKiYiIiRGK0YqRitGMUY7RjFGKy1JI1JzR0YoRilGK0YrKiZGO0YrSSNSMEdGKEYrRjFGMkYrRi1GK0Yx

u1:=collect(u0*(x^2+1),[Ss,R0,Rs],distributed,factor);

LDgqKEkkRGVzRzYiIiIiLCYqJClJInhHRiUiIiNGJkYmRiZGJkYmKS1JI1NzR0YlNiNGKkYrRiZGJiosIiIlRiZGJEYmRidGJkYtRiYtRi42IyokRiohIiJGJkY1KixGK0YmLCZGKkYmRiZGJkY1RiRGJkYnRiZGLUYmRjUqKkYxRiZGJEYmRidGJilGMkYrRiZGJiosRjFGJkY3RjVGJEYmRidGJkYyRiZGJiomLCYqJiIiKkYmRilGJkY1Rj5GNUYmKUkjUjBHRiVGK0YmRiYqLCIjPUYmRjdGJkYnRiZGQEYmLUkjUnNHRiVGL0YmRjUqKiIiJ0YmRjdGNUYnRiZGQEYmRiYqKkY+RiYpRjdGK0YmRidGJilGQ0YrRiZGNSomLCYqJkZGRiZGKUYmRiZGRkYmRiZGQ0YmRiYqKCwmRiRGJkYmRjVGJkYnRiZGSEY1RiY=

u1expl:=collect(subs(Des=Deltas,u0),[Ss,Rs,R0],distributed,factor);

LDgqKCwqKigiIiUiIiIpSSJ0RzYiIiIjRicpSSJ4R0YqRitGJ0YnKigiIilGJ0YoRidGLUYnRicqJkYmRidGKEYnRidGLSEiIkYnRi1GMSktSSNTc0dGKjYjRi1GK0YnRjEqLEYmRidGJEYnRi1GMUYzRictRjQ2IyokRi1GMUYnRicqLEYrRidGJEYnLCZGLUYnRidGJ0YxRi1GMUYzRidGJyoqRiZGJ0YkRidGLUYxKUY3RitGJ0YxKixGJkYnRiRGJ0Y7RjFGLUYxRjdGJ0YxKigiIipGJylGO0YrRicpLUkjUnNHRipGNUYrRidGMSooLCYqJiIjPUYnRi1GJ0YxRkhGMUYnSSNSMEdGKkYnRkNGJ0YnKiYiIidGJ0ZDRidGJyomRkBGJylGSUYrRidGMSooRktGJ0Y7RjFGSUYnRicqKEYmRidGKEYnRi1GMUYx

series(u1expl,x=-1,1);

KycsJkkieEc2IiIiIkYmRiYsJiomIiIjRiYtSSNTc0dGJTYjISIiRiZGJiomIiInRiZJI1IwR0YlRiZGJkYtLUkiT0clKnByb3RlY3RlZEc2I0YmIiIh

R0sol:=-Ss(-1)/3:

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

u2:=collect(u1expl-convert(%%,polynom),[Ss,Rs,R0],distributed);

LDgqKCIiIyIiIiwmSSJ4RzYiRiVGJUYlISIiLUkjU3NHRig2I0YpRiVGKSooLCoqKCIiJUYlKUkidEdGKEYkRiUpRidGJEYlRiUqKCIiKUYlRjFGJUYnRiVGJSomRjBGJUYxRiVGJUYnRilGJUYnRikpLUYrNiNGJ0YkRiVGKSosRjBGJUYuRiVGJ0YpRjhGJS1GKzYjKiRGJ0YpRiVGJSosRiRGJUYuRiVGJkYpRidGKUY4RiVGJSoqRjBGJUYuRiVGJ0YpKUY7RiRGJUYpKixGMEYlRi5GJUYmRilGJ0YpRjtGJUYpKigiIipGJSlGJkYkRiUpLUkjUnNHRihGOUYkRiVGKSooLCYqJiIjPUYlRidGJUYpRktGKUYlSSNSMEdGKEYlRkZGJUYlKiYiIidGJUZGRiVGJSomRkNGJSlGTEYkRiVGKSooRjBGJUYxRiVGJ0YpRik=

(note that Ss(0)=0)

Ssser(3);

LCgqKEkieEc2IiIiIiwoKiYiIzVGJilGJCIiI0YmRiYqJiIjV0YmRiRGJkYmIiNsRiZGJilJInRHRiUiIidGJkYmKihGJEYmLCYqJiIiJEYmRiRGJkYmIiIoRiZGJilGMCIiJUYmRiYqJilGMEYrRiZGJEYmRiY=

We must be a bit careful with the terms in 1/(x+1)

u2n:=op(5,u2)+op(6,u2)-4*Ss(-1)/(x+1);

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series(u2n,x=-1,2);

KycsJkkieEc2IiIiIkYmRiYsJiomIiIlRiYpLUkjU3NHRiU2IyEiIiIiI0YmRiYqJkYpRiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRixGLUYmRi4iIiEtSSJPR0Y1NiNGJkYm

u2p:=subs(Ss(1/x)=0,u2-u2n);

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Terms with no series in 1/x

subs(Ss(0)=0,series(u2p,x,1));

KylJInhHNiIsJComIiIlIiIiKUkidEdGJCIiI0YoISIiRiwsLiomRitGKC1JI1NzR0YkNiNGLEYoRigqKCIiKUYoRilGKC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGMDYjIiIhRihGKComIiIqRigpLUkjUnNHRiRGO0YrRihGLCooIiM9RihJI1IwR0YkRihGQEYoRiwqJiIiJ0YoRkBGKEYoKiZGPkYoKUZERitGKEYsRjwtSSJPR0Y4NiNGKEYo

The parts with S(1/x) are x-negative

factor(subs(Ss(0)=0,series(subs(x=1/x,u2n),x,3)));

KydJInhHNiIsKCooIiM7IiIiKUkidEdGJCIiI0YoKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJI1NzR0YkNiMiIiFGK0YoISIiKihGJ0YoRilGKEYtRihGNyomIiIlRigtRjQ2I0Y3RihGN0YoLUkiT0dGMTYjRihGKw==

The negative part: a bit simpler than in the square lattice case

u2neg:=-4*Sneg(1/x)-4*t^2/x+u2n;

LCwqJiIiJSIiIi1JJVNuZWdHNiI2IyokSSJ4R0YoISIiRiVGLCooRiRGJSlJInRHRigiIiNGJUYrRixGLCoqRiRGJSwqKihGJEYlRi5GJSlGK0YwRiVGJSooIiIpRiVGLkYlRitGJUYlKiZGJEYlRi5GJUYlRitGLEYlRitGLCktSSNTc0dGKEYpRjBGJUYsKixGJEYlRjJGJSwmRitGJUYlRiVGLEYrRixGOUYlRiwqKEYkRiVGPEYsLUY6NiNGLEYlRiw=

Dividing by 4

u2neg:=-Sneg(1/x)-t^2/x+u2n/4;

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eqpap:=Deltas*(Ss(1/x)^2+Ss(1/x)/(x+1))-Sneg(1/x)-Ss(-1)/(x+1)-t^2/x;

LCoqJiwmIiIiRiUqKiIiJUYlKSwmSSJ4RzYiRiVGJUYlIiIjRiVGKiEiIilJInRHRitGLEYlRi1GJSwmKiQpLUkjU3NHRis2IyokRipGLUYsRiVGJSomRjNGJUYpRi1GJUYlRiUtSSVTbmVnR0YrRjVGLSomRilGLS1GNDYjRi1GJUYtKiZGLkYlRipGLUYt

factor(eqpap/u2neg);

IiIi

series(u2neg,x=-1,2);

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let us check: this must be non-negative

n:=10: factor(series(subs(Sneg(1/x)=Deltas*Ss(x)*Ss(1/x),Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)) , Ss(-1)=subs(x=-1,Ssser(n)), u2neg),t,n));

KytJInRHNiIhIiIiIiUsJkkieEdGJCIiIiIiJ0YlRiosKComIiIjRikpRihGLUYpRikqJiIjOkYpRihGKUYpIiNfRiUiIiktSSJPRyUqcHJvdGVjdGVkRzYjRikiIzU=

and this must be negative

n:=10: expand(factor(series(subs(Sneg(1/x)=0,Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)) , Ss(-1)=subs(x=-1,Ssser(n)), u2neg),t,n)));

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

u5:=collect((factor( Des*Ss(x)*Ss(1/x)-solve(u2neg,Sneg(1/x))-P0-subs(x=1/x,solve(u2neg,Sneg(1/x)) ))),[Des,Ss,P0],factor);

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eqpap:=Des*(Ss(x)^2+Ss(1/x)^2-Ss(x)*Ss(1/x)+Ss(x)/(1/x+1)+Ss(1/x)/(x+1))-Ss(-1)+P0-t^2*(x+1/x);

LCoqJkkkRGVzRzYiIiIiLCwqJCktSSNTc0dGJTYjSSJ4R0YlIiIjRiZGJiokKS1GKzYjKiRGLSEiIkYuRiZGJiomRipGJkYxRiZGNComRipGJiwmRjNGJkYmRiZGNEYmKiZGMUYmLCZGLUYmRiZGJkY0RiZGJkYmLUYrNiNGNEY0SSNQMEdGJUYmKiYpSSJ0R0YlRi5GJiwmRi1GJkYzRiZGJkY0

factor(subs(Des=Deltas,u5+eqpap));

IiIh

Let us check

n:=15: factor(series(subs(Des=Deltas,Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)), P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0), Ss(-1)=subs(x=-1,Ssser(n)), u5),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

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

tc:=subs(Des=0,eqpap);

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with(gfun):Drac:=algeqtoseries(numer(Deltas),t,x,5);

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normal(series(subs(x=Drac[1],P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0), Ss(-1)=subs(x=-1,Ssser(n)),tc),t,5));

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normal(series(subs(x=Drac[2],P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0), Ss(-1)=subs(x=-1,Ssser(n)),tc),t,5));

KylJInRHNiIjISIiIiIlIiIhIiIkIiIjLUkiT0clKnByb3RlY3RlZEc2IyIiIkYn

What to remember from the previous sub-section

f13s:=(Ss(x)-2*Ss(1/x)-1/(x+1))*sqrt(Des)+(-3*x-3)*Rs(x)-3*R0+1/(x+1);

LCoqJiwoLUkjU3NHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwKiQsJkYpRipGKkYqRjBGMEYqKUkkRGVzR0YnI0YqRixGKkYqKiYsJiomIiIkRipGKUYqRjBGOUYwRiotSSNSc0dGJ0YoRipGKiomRjlGKkkjUjBHRidGKkYwRjFGKg==

R0sol:=-Ss(-1)/3;

LCQqJiMiIiIiIiRGJS1JI1NzRzYiNiMhIiJGJUYr

u5form:=Des*(Ss(x)^2+Ss(1/x)^2-Ss(x)*Ss(1/x)+Ss(x)/(1/x+1)+Ss(1/x)/(x+1))-Ss(-1)+P0-t^2*(x+1/x);

Deltas:=collect(subs(x=sqrt(x),expand(Delta)),t,factor);

LCYiIiJGIyoqIiIlRiMpLCZJInhHNiJGI0YjRiMiIiNGI0YoISIiKUkidEdGKUYqRiNGKw==

u5:=subs(Des=Deltas,u5form);

SSd1NWZvcm1HNiI=

Rser:=proc(n) collect(t^2/x*subs(y=0,Mser(n-1)),t,factor): end:

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

Rsser:=proc(n) subs(x=sqrt(x),Rser(n)): end:

Ssser:=proc(n) subs(x=sqrt(x),Sser(n)): end:

Ssser(5);Rsser(5);

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LCgqJiwoKiYjIiIjIiIkIiIiKUkieEc2IkYnRilGKSomIiIlRilGK0YpRikjIiNKRihGKUYpKUkidEdGLCIiJ0YpRikqJiwmI0YuRihGKSomI0YpRihGKUYrRilGKUYpKUYyRi5GKUYpKiZGOEYpKUYyRidGKUYp

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

La multiplication qu'il faut

u11:=collect(expand(u5form*(1+Ss(x)+Ss(1/x))),[Des,Ss,P0],distributed,factor);

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

IiIi

factor(series(subs(Des=Deltas,u11),x=-1,2));

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Extract the non-negative part, being careful with the divisions by 1/(1+x)

u11p:=subs(Ss(1/x)=0,u11)+Ss(-1)*(Ss(-1)+1)/(x+1);

LDYqKCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiopSSJ0R0YoRilGKkYnISIiRi0qKkYkRipGK0YqRidGLS1JI1NzR0YoNiNGJ0YqRi1JI1AwR0YoRioqJkYyRipGL0YqRioqKiwmKiZGKUYqRidGKkYqRipGKkYqLCZGJ0YqRipGKkYtSSREZXNHRihGKilGL0YpRipGKiomRjhGKilGLyIiJEYqRioqJi1GMDYjRi1GKkYvRipGLSoqRidGKkY3Ri1GL0YqRjhGKkYqRj5GLSooRj5GKiwmRj5GKkYqRipGKkY3Ri1GKg==

factor(series(subs(Des=Deltas,u11p),x=-1,1));

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u11n:=collect(u11-u11p,[Des,Ss],factor);

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A part to be substracted

normal(subs(Ss(0)=0,series(subs(Des=Deltas,u11p),x,1)));

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factor(subs(Ss(0)=0,series(subs(x=1/x,Des=Deltas,u11n),x,2)));

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eqSp:=collect(u11p-convert(%%,polynom)+convert(%,polynom),[Des,Ss(x),x],factor);

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Let us check this equation

n:=10: normal(series(subs(Des=Deltas,D(Ss)(0)=coeff(Ssser(n),x,1),Ss(-1)=subs(x=-1,Ssser(n)),Ss(0)=subs(x=0,Ssser(n)),Ss(x)=Ssser(n),P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0),eqSp),t,n+1));

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Paper version

eqpap:=Des*(Ss(x)^3+(2*x+1)/(1+x)*Ss(x)^2+Ss(x)/(1/x+1))-(1+Ss(x))*(Ss(-1)-P0+t^2*(x+1/x))-t^2*Ss1+Ss(-1)*(1+Ss(-1))/(x+1)+t^2/x;

LCwqJkkkRGVzRzYiIiIiLCgqJCktSSNTc0dGJTYjSSJ4R0YlIiIkRiZGJiooLCYqJiIiI0YmRi1GJkYmRiZGJkYmLCZGLUYmRiZGJiEiIilGKkYyRiZGJiomRipGJiwmKiRGLUY0RiZGJkYmRjRGJkYmRiYqJiwmRiZGJkYqRiZGJiwoLUYrNiNGNEYmSSNQMEdGJUY0KiYpSSJ0R0YlRjJGJiwmRi1GJkY4RiZGJkYmRiZGNComRkBGJkkkU3MxR0YlRiZGNCooRjxGJiwmRjxGJkYmRiZGJkYzRjRGJiomRkBGJkYtRjRGJg==

factor(subs(Ss1=D(Ss)(0),eqSp-eqpap));

IiIh

Extract the constant term

normal(subs(Ss(0)=0,series(subs(Des=Deltas,eqSp),x=0,2)));

KylJInhHNiIsKCooIiIjIiIiKUkidEdGJEYnRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSNTc0dGJDYjIiIhRighIiIqJCktRjI2I0Y1RidGKEYoSSNQMEdGJEYoRjQsMiooIiIlRihGKUYoKUYrRidGKEY1KigjRihGJ0YoRilGKC0tLUkjQEBHRi42JEYtRidGMUYzRihGNSooRj1GKEYpRihGK0YoRjUqJkYrRihGOEYoRjUqJkYrRihGOkYoRigqJEYpRihGNUY4RjVGNkY1RigtSSJPR0YvNiNGKEYn

map(factor,subs(Ss(0)=0,convert(subs(Ss(x)=0,eqSp),parfrac,x)));

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eqSp1:=Des*(Ss(x)^3+(2*x+1)/(1+x)*Ss(x)^2+Ss(x)/(1/x+1))-(1+Ss(x))*(-F0+t^2*(x+1/x))-t^2*Ss1+(2*t^2*Ss1-F0)/(x+1)+t^2/x;;

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F0ser:=proc(n) series(coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0)-subs(x=-1,Ssser(n)),t,n+1): end:

F0ser(8);

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n:=7: normal(series(subs(Des=Deltas,Ss(x)=Ssser(n),Ss1=coeff(Ssser(n),x,1), F0=F0ser(n),eqSp1),t,n+1));

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series(subs(Des=Deltas,eqSp1),x=-1,2);

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factor(subs(Ss(0)=0,series(subs(Des=Deltas,eqSp1),x,2)));

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What needs to be remembered from this subsection

F0ser:=proc(n) series(coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0)-subs(x=-1,Ssser(n)),t,n+1): end:

F0ser(3);

KydJInRHNiIiIiIiIiMtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIiU=

eqSp1:=Des*(Ss(x)^3+(2*x+1)*Ss(x)^2/(x+1)+Ss(x)/(1/x+1))-(1+Ss(x))*(-F0+t^2*(x+1/x))-t^2*Ss1+(2*Ss1*t^2-F0)/(x+1)+t^2/x;

LCwqJkkkRGVzRzYiIiIiLCgqJCktSSNTc0dGJTYjSSJ4R0YlIiIkRiZGJiooLCYqJiIiI0YmRi1GJkYmRiZGJkYmKUYqRjJGJiwmRi1GJkYmRiYhIiJGJiomRipGJiwmKiRGLUY1RiZGJkYmRjVGJkYmRiYqJiwmRiZGJkYqRiZGJiwmSSNGMEdGJUY1KiYpSSJ0R0YlRjJGJiwmRi1GJkY4RiZGJkYmRiZGNSomRj5GJkkkU3MxR0YlRiZGNSomLCYqKEYyRiZGQkYmRj5GJkYmRjxGNUYmRjRGNUYmKiZGPkYmRi1GNUYm

n:=20: normal(series(subs(Des=Deltas,Ss(x)=Ssser(n),Ss1=coeff(Ssser(n),x,1), F0=F0ser(n),eqSp1),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNB

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

eqS:=map(factor,subs(Des=Deltas,Ss(x)=Ssx,eqSp1));

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Differentiating with respect to x_0

eqSS:=diff(eqS,Ssx);

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eqSSnum:=collect(x*(x+1)*eqSS,t,factor);

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Evaluation at t=0 : gives two roots X0 and X1, with CT 0 and first coefficient +/-1

subs(t=0,Ssx=0,F0=0,%);

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collect(subs(Ssx=t*x*F(x),x=t*z,F0=t^2*G0, eqSSnum/t^2),[Ssx,t],distributed,factor);

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subs(t=0,%);

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n:=10:expr:=normal(series(subs(Ssx=Ssser(n),F0=F0ser(n),eqSS*x*(x+1)),t,n+1));

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with(gfun):Xisol:=algeqtoseries(convert(expr,polynom),t,x,8,true);

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We conclude that eqSS has two roots X0 X1, both Catalan-ish

Let us determine the first coefficients

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

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series(X0rac,t,20);

K0lJInRHNiIiIiJGJSIiI0YmIiImIiIkIiM5IiIlIiNVRiciJEsiIiInIiRIJSIiKCIlSTkiIikiJWlbIiIqIiYnejsiIzUiJid5ZSIjNiInNyEzIyIjNyInK0h1IiM4IihTV24jRikiKFhbcCoiIzoiKXF3Tk4iIzsiKiF6VydIIiIjPCIqKyhRd1oiIz0tSSJPRyUqcHJvdGVjdGVkRzYjRiUiIz4=

X1rac:=-(1+2*t-sqrt(1+4*t))/2/t;

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Let us check this conjecture

n:=20: normal(series(subs(Ssx=Ssser(n), F0=F0ser(n),x=X0rac,eqSS),t,n+1));

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

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

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algX1:=subs(t=-t,X0=X1,algX0);

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Differentiating w.r.t. x

eqSx:=collect(diff(eqS,x),[Ssx,F0,Ss1],factor);

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n:=20: normal(series(subs(Ssx=Ssser(n), Ss1=coeff(Ssser(n),x,1),F0=F0ser(n),x=X0rac,eqSx),t,n+1));

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

factor(coeff(eqS,F0));factor(coeff(eqSS,F0));factor(coeff(eqSx,F0));

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IiIi

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We eliminate F0

collect(factor(eqS-(Ssx*x+Ssx+x)/(x+1)*eqSS),Ss1,factor);

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collect(factor(eqSx-1/(x+1)^2*eqSS),Ss1,factor);

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and now eliminate Ss1

factor(%*(x-1)-2*%%/(x+1));

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The complete expression

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

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The explanation for X0

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

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expand(subs(X0=x,algX0));

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Why X0 is a solution:

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

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

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

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op(1,e1);

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algX0;

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The term that vanishes

e2:=factor(op(2,e1));

LGJ3KioiJDMiIiIiKUkkU3MxRzYiIiIjRiUpSSJ0R0YoIiInRiUpSSJ4R0YoIiIoRiUhIiIqKiIiKUYlRidGJUYqRiUpRi5GMkYlRjAqKiIkOyNGJUYmRiVGKkYlKUYuRixGJUYwKioiI3NGJUYnRiVGKkYlRi1GJUYwKixGOEYlSSNGMEdGKEYlRidGJSlGKyIiJUYlRi1GJUYlKipGPEYlRjpGJUY7RiVGM0YlRiUqKkYkRiVGJkYlRipGJSlGLiIiJkYlRiUqKiIkNyJGJUYnRiVGKkYlRjZGJUYwKihGPEYlRipGJUYtRiVGJSosIiRXIkYlRjpGJUYnRiVGO0YlRjZGJUYlKioiI1NGJUY6RiVGO0YlRi1GJUYlKioiJEslRiVGJkYlRipGJSlGLkY8RiVGJSoqIiNGRiVGJkYlRjtGJUY2RiVGJSoqRjhGJUYnRiVGKkYlRj9GJUYlKioiIzlGJUYnRiVGO0YlRi1GJUYwKihGR0YlRipGJUY2RiVGJSomRjtGJUYzRiVGMCoqRjJGJSlGOkYpRiUpRitGKUYlRi1GJUYwKixGOEYlRjpGJUYnRiVGO0YlRj9GJUYwKipGQkYlRjpGJUY7RiVGNkYlRiUqKkYkRiVGJkYlRipGJSlGLiIiJEYlRiUqKiIkUyNGJUYnRiVGKkYlRkpGJUYlKioiI0VGJUYnRiVGO0YlRjZGJUYwKigiJEMiRiVGKkYlRj9GJUYlKigiIzVGJUY7RiVGLUYlRjAqKkYyRiVGU0YlRlRGJUY2RiVGJSosIiQpR0YlRjpGJUYnRiVGO0YlRkpGJUYwKiwiIz1GJUY6RiVGJ0YlRlRGJUY2RiVGMCoqIiRfIkYlRjpGJUY7RiVGP0YlRiUqKkYpRiVGOkYlRlRGJUYtRiVGJSoqRjVGJUYmRiVGKkYlKUYuRilGJUYwKioiI2FGJUYmRiVGO0YlRkpGJUYwKipGOEYlRidGJUYqRiVGWEYlRiUqKkZPRiVGJ0YlRjtGJUY/RiVGJSooIiR3IkYlRipGJUZKRiVGJSooIiNIRiVGO0YlRjZGJUYwKihGPEYlKUY6RllGJUY2RiVGJSoqIiQvIkYlRlNGJUZURiVGP0YlRiUqLEY4RiVGOkYlRidGJUY7RiVGWEYlRjAqKkZib0YlRjpGJUY7RiVGSkYlRiUqKkZbb0YlRjpGJUZURiVGNkYlRjAqKkYkRiVGJkYlRipGJUYuRiVGMCoqRkJGJUYnRiVGKkYlRmVvRiVGMCoqIiNfRiVGJ0YlRjtGJUZKRiVGJSoqRjxGJUYnRiVGVEYlRjZGJUYlKihGaW5GJUYqRiVGWEYlRiUqKCIjUUYlRjtGJUY/RiVGMCooIiM7RiVGX3BGJUY/RiVGJSoqRltwRiVGU0YlRlRGJUZKRiVGJSomRlNGJUY2RiVGMCosRkVGJUY6RiVGJ0YlRjtGJUZlb0YlRiUqLCIjT0YlRjpGJUYnRiVGVEYlRkpGJUYlKipGYm9GJUY6RiVGO0YlRlhGJUYlKioiI11GJUY6RiVGVEYlRj9GJUYwKipGTEYlRiZGJUY7RiVGZW9GJUYlKipGOEYlRidGJUYqRiVGLkYlRjAqKkZPRiVGJ0YlRjtGJUZYRiVGJSoqRilGJUYnRiVGVEYlRj9GJUYlKihGR0YlRipGJUZlb0YlRiUqKEZjcUYlRjtGJUZKRiVGMCooRilGJUZURiVGNkYlRiUqKCIjQ0YlRl9wRiVGSkYlRiUqKkZhcEYlRlNGJUZURiVGWEYlRiUqKCIjN0YlRlNGJUY/RiVGMCosRjhGJUY6RiVGJ0YlRjtGJUYuRiVGJSoqRkJGJUY6RiVGO0YlRmVvRiVGJSoqIiN3RiVGOkYlRlRGJUZKRiVGMCooRjJGJUYnRiVGKkYlRjAqKkZnbkYlRidGJUY7RiVGZW9GJUYwKipGYnJGJUYnRiVGVEYlRkpGJUYwKihGPEYlRipGJUYuRiVGJSooRlxxRiVGO0YlRlhGJUYwKihGLEYlRlRGJUY/RiVGJSooRl5xRiVGX3BGJUZYRiVGJSoqRjJGJUZTRiVGVEYlRmVvRiVGJSooIiNBRiVGU0YlRkpGJUYwKixGYG9GJUY6RiVGJ0YlRlRGJUZlb0YlRjAqKkZHRiVGOkYlRjtGJUYuRiVGJSoqRmZxRiVGOkYlRlRGJUZYRiVGMCooRilGJUY6RiVGP0YlRiUqKkZPRiVGJ0YlRjtGJUYuRiVGMCoqRilGJUYnRiVGVEYlRlhGJUYlKihGXXBGJUY7RiVGZW9GJUYwKihGMkYlRlRGJUZKRiVGJSooRjxGJUZfcEYlRmVvRiVGJSoqRjJGJUZTRiVGVEYlRi5GJUYwKihGYnJGJUZTRiVGWEYlRjAqKEY8RiVGOkYlRjtGJUYlKipGW29GJUY6RiVGVEYlRmVvRiVGMCooRjJGJUY6RiVGSkYlRiUqKkY8RiVGJ0YlRlRGJUZlb0YlRiUqKEZbb0YlRjtGJUYuRiVGMCooRixGJUZURiVGWEYlRiUqJkZTRiVGZW9GJUYwKipGKUYlRjpGJUZURiVGLkYlRiUqKEYpRiVGOkYlRlhGJUYlKiRGO0YlRjAqKEYpRiVGVEYlRmVvRiVGJSokRkpGJUYw

and then of x

e3:=(factor(resultant(subs(x=X0,e2),algX0,X0)));

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

A first relation between Sp0 and F0

eqF0Ss11:=op(1,op(2,e3));

LFAqKCIkSyUiIiIpSSRTczFHNiIiIiNGJSlJInRHRigiIidGJUYlKigiJDsjRiVGJkYlKUYrIiImRiUhIiIqKCIkZyJGJUYnRiVGKkYlRiUqKiIkKUdGJUkjRjBHRihGJUYnRiUpRisiIiVGJUYxKigiI0ZGJUYmRiVGN0YlRiUqKCIjO0YlRidGJUYvRiVGMSomRjxGJUYqRiVGJSoqIiRXIkYlRjZGJUYnRiUpRisiIiRGJUYlKigiI1tGJUY2RiVGN0YlRjEqKCIjVUYlRidGJUY3RiVGJSomIiIpRiVGL0YlRiUqKCIjY0YlKUY2RilGJSlGK0YpRiVGJSoqIiM9RiVGNkYlRidGJUZLRiVGMSooRjxGJUY2RiVGQEYlRiUqKCIjR0YlRidGJUZARiVGMSomIiM2RiVGN0YlRiUqJkY4RiUpRjZGQUYlRiUqKEY8RiVGSkYlRitGJUYxKigiIzlGJUY2RiVGS0YlRjEqKEY4RiVGJ0YlRktGJUYlKiZGOEYlRkBGJUYxKiRGSkYlRjEqKEY4RiVGNkYlRitGJUYl

Let us check

n:=10: normal(series(subs(Ssx=Ssser(n),F0=F0ser(n),Ss1=coeff(Ssser(n),x,1),eqF0Ss11),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

We now work with X1, that is, replace t by -t

e4:=(factor(resultant(subs(x=X1,e2),algX1,X1)));

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

A second relation between Ss1 and F0

eqF0Ss12:=op(1,op(2,e4));

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

Let us check

n:=10: normal(series(subs(Ssx=Ssser(n),F0=F0ser(n),Ss1=coeff(Ssser(n),x,1),eqF0Ss12),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

Finally we eliminate F0 to get an equation for Ss1

e7:=factor(resultant(eqF0Ss11,eqF0Ss12,F0));

LCQqKiInV0BFIiIiKUkidEc2IiIiKkYlKSwmSSRTczFHRihGJUYlRiUiIiNGJSw8KigiJiRvPkYlKUYsIiIlRiUpRiciIidGJUYlKigiJlNQJUYlKUYsIiIkRiVGM0YlRiUqKCImZVEkRiUpRixGLUYlRjNGJUYlKigiJSg9I0YlRjdGJSlGJ0YyRiUhIiIqKCImSzYiRiVGLEYlRjNGJUYlKigiJXBSRiVGO0YlRj5GJUY/KiYiJUo4RiVGM0YlRiUqKCIlWEBGJUYsRiVGPkYlRj8qKCIjIilGJUY7RiUpRidGLUYlRiUqJiIkMiJGJUY+RiVGPyooIiQ5IkYlRixGJUZKRiVGJSokRkpGJUYlRixGP0YlRiU=

The same equation as in the ordinary square lattice!

algSs1:=collect(op(4,e7),Ss1,factor);

LCwqKCImJG8+IiIiKUkkU3MxRzYiIiIlRiUpSSJ0R0YoIiInRiVGJSoqIiUoPSNGJSlGK0YpRiUsJiomIiM/RiUpRisiIiNGJUYlRiUhIiJGJSlGJyIiJEYlRiUqLCIjIilGJUYzRiUsJiomIiM2RiVGM0YlRiVGJUY1RiUsJiomIiNRRiVGM0YlRiVGJUY1RiUpRidGNEYlRiUqKCwmKiYiIyMqRiVGM0YlRiVGJUY1RiUpRjpGNEYlRidGJUYlKiZGM0YlLCgqJiIlSjhGJUYvRiVGJSomIiQyIkYlRjNGJUY1RiVGJUYlRiU=

And now we eliminate Ss1 to get an equation for F0

e8:=factor(resultant(eqF0Ss11,eqF0Ss12,Ss1));

LCQqKiIlN3AiIiIpSSJ0RzYiIiM1RiUsPiomIiNbRiUpRiciIilGJUYlKigiJEMjRiVJI0YwR0YoRiUpRiciIidGJUYlKigiJGckRiUpRjEiIiNGJSlGJyIiJUYlRiUqJiIjKSlGJUYyRiVGJSooIiQ7I0YlKUYxIiIkRiUpRidGN0YlRiUqKCIjb0YlRjFGJUY4RiUhIiIqJiIjRkYlKUYxRjlGJUYlKigiJEUiRiVGNkYlRkBGJUZDKiYiIz9GJUY4RiVGQyomRkVGJUY+RiVGQyooIiM7RiVGMUYlRkBGJUYlKiYiIipGJUY2RiVGJSokRkBGJUYlRjFGQ0YlKSwoKiZGLkYlRkBGJUYlKiZGOUYlRjFGJUYlRiVGQ0Y3RiVGJQ==

algF0:=map(factor,collect(op(3,e8),F0,factor));

LCwqJiIjRiIiIilJI0YwRzYiIiIlRiVGJSooRiRGJSwmKiYiIilGJSlJInRHRigiIiNGJUYlRiUhIiJGJSlGJyIiJEYlRiUqLCIiKkYlLCYqJkYwRiVGL0YlRiVGJUYlRiUsJkY3RiVGJUYxRiUsJiomIiM1RiVGLkYlRiVGJUYxRiUpRidGMEYlRiUqJiwqKiYiJEMjRiUpRi8iIidGJUYlKiYiI29GJSlGL0YpRiVGMSomIiM7RiVGLkYlRiVGJUYxRiVGJ0YlRiUqJkYuRiUsKiomIiNbRiVGQUYlRiUqJiIjKSlGJUZFRiVGJSomIiM/RiVGLkYlRjFGJUYlRiVGJQ==

The equations for F0 and Ss1

algSs1;

algF0 := map(factor,27*F0^4+(216*t^2-27)*F0^3+(9*(2*t+1))*(2*t-1)*(10*t^2-1)*F0^2+(224*t^6-68*t^4+16*t^2-1)*F0+t^2*(48*t^6+88*t^4-20*t^2+1));

<Text-field style="Heading 2" layout="Heading 2">Parametrization for F0 and Ss1</Text-field>

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

Both series F0 and Ss1 are even in t

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

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpRigiIiRGKiksJkYoRiVGLEYlRixGJUYl

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(t=sqrt(t2Tm),algF0)));

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

n:=6:normal(series(subs(Tm=Tmser(n),F0=F0ser(n),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),F0=F0ser(n),op(2,res)),t,n+1));

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KytJInRHNiIhKF9yNCMiIiEhKC9WPiUiIiMiKXdIRUYiIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

F0Tm:=factor(solve(op(1,res),F0));

LCQqKiMiIiIiJEciRiUsJkkjVG1HNiJGJUYlISIiRiUsKiomIiIkRiUpRihGLUYlRiUqJiIjSEYlKUYoIiIjRiVGKiomIiM6RiVGKEYlRioiIipGJUYlRi5GKkYq

res:=numer(factor(subs(t=sqrt(t2Tm),algSs1)));

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

n:=6:normal(series(subs(Tm=Tmser(n),Ss1=coeff(Ssser(n),x,1),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),Ss1=coeff(Ssser(n),x,1),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIhJ1dARSIiISIpa09CQCIiIyIqXzZDayMiIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

Ss1Tm:=factor(solve(op(1,res),Ss1));

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

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==

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

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

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

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

F0Tm:=-(1/128)*(Tm-1)*(3*Tm^3-29*Tm^2-15*Tm+9)/Tm^3; Ss1Tm:=-(Tm-1)*(Tm^2-6*Tm-11)/(Tm+3)^3;

LCQqKiMiIiIiJEciRiUsJkkjVG1HNiJGJUYlISIiRiUsKiomIiIkRiUpRihGLUYlRiUqJiIjSEYlKUYoIiIjRiVGKiomIiM6RiVGKEYlRioiIipGJUYlRi5GKkYq

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

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

eqS;

LCwqLiwqKigiIiUiIiIpSSJ0RzYiIiIjRicpSSJ4R0YqRitGJ0YnKigiIilGJ0YoRidGLUYnRicqJkYmRidGKEYnRidGLSEiIkYnSSRTc3hHRipGJywmRidGJ0YyRidGJywoKiZGMkYnRi1GJ0YnRjJGJ0YtRidGJywmRi1GJ0YnRidGMUYtRjFGMSooRjNGJywoKiZGKEYnRixGJ0YxKiZJI0YwR0YqRidGLUYnRicqJEYoRidGMUYnRi1GMUYnKiZGKEYnSSRTczFHRipGJ0YxKiYsJiooRitGJ0Y+RidGKEYnRjFGO0YnRidGNkYxRjEqJkYoRidGLUYxRic=

Ss(x) is even in t

Ssser(4);

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

algSsTm0:=collect(numer(factor(subs(F0=F0Tm,Ss1=Ss1Tm,eqS))),Ssx,factor);

LCoqLiIkRyIiIiIpSSNUbUc2IiIiJEYlLCoqKCIiJUYlKUkidEdGKCIiI0YlKUkieEdGKEYvRiVGJSooIiIpRiVGLUYlRjFGJUYlKiZGLEYlRi1GJUYlRjEhIiJGJSwmRjFGJUYlRiVGJSksJkYnRiVGKUYlRilGJSlJJFNzeEdGKEYpRiVGNSouRiRGJUYmRiUsJiomRi9GJUYxRiVGJUYlRiVGJUYqRiVGN0YlKUY6Ri9GJUY1KihGN0YlLD4qKiIkUydGJUYmRiVGLUYlKUYxRilGJUYlKioiJV82RiVGJkYlRi1GJUYwRiVGJSooRilGJSlGJ0YsRiVGMEYlRiUqKkZCRiVGJkYlRi1GJUYxRiVGJSooRilGJUZHRiVGMUYlRiUqKEYkRiVGJkYlRi1GJUYlKigiJGciRiVGJkYlRjBGJUY1KigiI0tGJUYmRiVGMUYlRjUqKCIjOUYlKUYnRi9GJUYwRiVGJSooRlBGJUZRRiVGMUYlRiUqKCIjQ0YlRidGJUYwRiVGJSooRlRGJUYnRiVGMUYlRiUqJiIiKkYlRjBGJUY1KiZGV0YlRjFGJUY1RiVGOkYlRjUqJkYxRiUsSCoqRiRGJSlGJyIiJ0YlRi1GJUYwRiVGJSoqRkVGJSlGJyIiJkYlRi1GJUYwRiVGJSooRilGJSlGJyIiKEYlRjFGJUYlKihGJEYlRmZuRiVGLUYlRiUqKiIlWz9GJUZpbkYlRi1GJUYxRiVGJSoqIiVjTUYlRkdGJUYtRiVGMEYlRiUqKEZqbkYlRmZuRiVGMUYlRjUqKCIkJyopRiVGaW5GJUYtRiVGNSoqIiUnNCVGJUZHRiVGLUYlRjFGJUYlKipGYm9GJUYmRiVGLUYlRjBGJUYlKigiJCQ+RiVGaW5GJUYxRiVGNSooRkJGJUZHRiVGLUYlRjUqKkZgb0YlRiZGJUYtRiVGMUYlRiUqKCIkTCdGJUZHRiVGMUYlRjUqKCIlMzlGJUYmRiVGLUYlRiUqKCIkeiNGJUYmRiVGMUYlRjUqKCIkWCpGJUZRRiVGMUYlRiUqKCIkMCVGJUYnRiVGMUYlRiUqJiIkViNGJUYxRiVGNUYlRjU=

We express this in terms of Tm only

algSsTm:=collect(op(3,factor(subs(t=sqrt(t2Tm),algSTm0))),Ssx,factor);

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

n:=10: normal(series(subs(Tm=Tmser(n),Ssx=Ssser(n),algSsTm),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

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

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

IiRkJg==

degree(res,Ssx);

IiM3

Parametrisation of x (from the Gessel paper)

xTmVm:=(1-Tm+3*Vm+Vm*Tm)/Vm^2/(3+Vm+Tm-Vm*Tm);

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Expansion of Vm in terms of (Tm-1)

with(gfun): VmserTm:=proc(n) series(Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Vm=Tm1*VmsTm1,numer(x-xTmVm)), Tm1,VmsTm1,n,true)),Tm1,n+2); end:

VmserTm(6);

KzFJJFRtMUc2IiMiIiIiIiVGJiwmKiYjRiYiIztGJkkieEdGJEYmRiZGKiEiIiIiIywoKiYjRiYiI0tGJilGLEYuRiZGJiomRjFGJkYsRiZGLSNGJiIja0YmIiIkLCoqJiNGJiIkRyJGJkYsRiZGJiomI0Y3RjtGJkYzRiZGLSomIyIiJiIkYyNGJilGLEY3RiZGJiNGJkZBRi1GJywqKiYjIiIoIiVDNUYmRjNGJkYmRj5GLSomI0ZHIiQ3JkYmKUYsRidGJkYmI0YmRkhGJkZALC4jRiYiJSc0JUYtKiYjIiNIRlBGJkZCRiZGJiomIyIjTiIlWz9GJkZMRiZGLSomI0ZARlBGJkYsRiZGLSomRllGJkYzRiZGJiomIyIjQEZXRiYpRixGQEYmRiYiIictSSJPRyUqcHJvdGVjdGVkRzYjRiZGRw==

Let us check

n:=9:normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),numer(x-xTmVm)),t,n+1));

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Let us prove that Vm parametrises the solution

res:=numer(factor(subs(x=xTmVm,algSsTm))):nops(%);indets(%%);

IiIj

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

IiIi

IiIj

We check that the quadratic term does not vanish

n:=10: normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Ssx=Ssser(n),op(1,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=14: normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Ssx=Ssser(n),op(2,res)),t,n+1));

KylJInRHNiIsJComIiVDNSIiIilJInhHRiQiIiNGKEYoIiM1LCoqJiIlV2hGKClGKiIiJEYoRigqJiImL3IlRihGKUYoRigqJiIlJzQlRihGKkYoISIiRidGNiIjNy1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjOQ==

SsTmVm:=factor(solve(op(1,res),Ssx));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJI1ZtR0YqRiVGJUYpISIiKiYiIiRGJUYrRiVGJUYlRiVGJSwuKiZGKUYlKUYrRiZGJUYlKihGJkYlRilGJUYrRiVGJSokRjFGJUYsRilGLComRiZGJUYrRiVGLEYuRixGJSwyKiYpRilGJkYlRjFGJUYlKihGJkYlRjdGJUYrRiVGLCooRiZGJUYpRiVGMUYlRiUqJEY3RiVGJSomRi5GJUYxRiVGLComRiZGJUYpRiVGJSomIiInRiVGK0YlRixGLkYsRiwsJkYrRiVGJUYlRixGLA==

Let's check

n:=18: normal(series(subs(Vm=convert(VmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),SsTmVm),t,n+1));

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

Ssser(16);

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

normal(series(%%-%,t,17));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM9

Expression of txM(0,x) (Vmd is Vm for x^2 rather than x)

M0xftxTmVm:=subs(Vm=Vmd,SsTmVm);

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsLiomRilGJSlGK0YmRiVGJSooRiZGJUYpRiVGK0YlRiUqJEYxRiVGLEYpRiwqJkYmRiVGK0YlRixGLkYsRiUsMiomKUYpRiZGJUYxRiVGJSooRiZGJUY3RiVGK0YlRiwqKEYmRiVGKUYlRjFGJUYlKiRGN0YlRiUqJkYuRiVGMUYlRiwqJkYmRiVGKUYlRiUqJiIiJ0YlRitGJUYsRi5GLEYsLCZGK0YlRiVGJUYsRiw=

let us check

n:=12: normal(series(subs(Vmd=subs(x=x^2,VmserTm(n)),Tm1=Tm-1,Tm=Tmser(n),M0xftxTmVm),t,n+1));

Ky1JInRHNiIqJClJInhHRiQiIiMiIiJGKCwmKiYiIiRGKSlGJyIiJUYpRikqJiIiKEYpRiZGKUYpRi4sKComIiM1RikpRiciIidGKUYpKiYiI1dGKUYtRilGKSomIiNsRilGJkYpRilGNSwqKiYiI05GKSlGJyIiKUYpRikqJiIkRCNGKUY0RilGKSomIiQiZUYpRi1GKUYpKiYiJCNwRilGJkYpRilGPi1JIk9HJSpwcm90ZWN0ZWRHNiNGKUYz

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

Ky1JInRHNiIqJClJInhHRiQiIiMiIiJGKComRiZGKSwmKiYiIiRGKUYmRilGKSIiKEYpRikiIiUqJkYmRiksKComIiM1RikpRidGL0YpRikqJiIjV0YpRiZGKUYpIiNsRilGKSIiJyomRiZGKSwqKiYiI05GKSlGJ0Y4RilGKSomIiREI0YpRjRGKUYpKiYiJCJlRilGJkYpRikiJCNwRilGKSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGKUYz

When x=1, the degree of Ss(1) decreases

factor(subs(x=1,algSTm));

LCQqLCIiKSIiIiwmSSNUbUc2IkYlRiVGJUYlKSwmRidGJSIiJCEiIiIiI0YlLCZGJUYlKiZGLUYlSSRTc3hHRihGJUYlRiUsLiooRi1GJSlGMEYtRiVGJ0YlRiUqJiIiJ0YlRjNGJUYsKihGLUYlRjBGJUYnRiVGJSomRjVGJUYwRiVGLEYnRiVGJUYsRiVGJQ==

factor(subs(x=1,Tm=Zm^2,algSTm));

LCQqLCIiKSIiIiwmKiQpSSNabUc2IiIiI0YlRiVGJUYlRiUpLCZGJ0YlIiIkISIiRitGJSwmRiVGJSomRitGJUkkU3N4R0YqRiVGJUYlLC4qKEYrRiUpRjJGK0YlRihGJUYlKihGK0YlRjJGJUYoRiVGJSomIiInRiVGNUYlRi9GJ0YlKiZGOEYlRjJGJUYvRiVGL0YlRiU=

factor([solve(2*Ssx^2*Tm-6*Ssx^2+2*Ssx*Tm-6*Ssx+Tm-1,Ssx)]);

NyQsJCooIyIiIiIiI0YmLChJI1RtRzYiISIiIiIkRiYqJCksJComLCZGKUYmRiZGJkYmLCZGKUYmRixGK0YmRitGJUYmRiZGJkYyRitGJiwkKihGJUYmLChGKUYmRixGK0YtRiZGJkYyRitGKw==

factor(subs(x=-11,algSTm));

LEYqKCIlK1MiIiIpSSRTc3hHNiIiIiRGJSlJI1RtR0YoIiIlRiUhIiIqKCImZywnRiVGJkYlKUYrRilGJUYtKigiJSslKUYlKUYnIiIjRiVGKkYlRi0qKCImKz8oRiVGJkYlKUYrRjRGJUYtKigiJ09qN0YlRjNGJUYwRiVGLSooIiVnXEYlRidGJUYqRiVGLSooIicrNzpGJUYzRiVGN0YlRi0qKCImd0gpRiVGJ0YlRjBGJUYtKiYiJDsnRiVGKkYlRi0qJiInKyEzIkYlRiZGJUYlKigiJiEzKSpGJUYnRiVGN0YlRi0qJiImb2oiRiVGMEYlRi0qJiInK29BRiVGM0YlRiUqKCIlIUcmRiVGJ0YlRitGJUYlKiYiJmMnPUYlRjdGJUYtKiYiJ2coXCJGJUYnRiVGJSomIiUncCRGJUYrRiVGJSImVz4kRiU=

algSsTm:=(4*(x+1))*(Tm^4*x^2+2*Tm^4*x+8*Tm^3*x^2+Tm^4-48*Tm^3*x+18*Tm^2*x^2+8*Tm^3+36*Tm^2*x+18*Tm^2-27*x^2-54*x-27)*Ssx^3+(4*(2*x+1))*(Tm^4*x^2+2*Tm^4*x+8*Tm^3*x^2+Tm^4-48*Tm^3*x+18*Tm^2*x^2+8*Tm^3+36*Tm^2*x+18*Tm^2-27*x^2-54*x-27)*Ssx^2+(5*Tm^4*x^3+15*Tm^4*x^2+40*Tm^3*x^3+11*Tm^4*x-248*Tm^3*x^2+90*Tm^2*x^3+Tm^4-24*Tm^3*x+190*Tm^2*x^2+8*Tm^3+118*Tm^2*x+48*Tm*x^2-135*x^3+18*Tm^2+48*Tm*x-261*x^2-153*x-27)*Ssx+x*(Tm-1)*(Tm^3*x^2+6*Tm^3*x+9*Tm^2*x^2+Tm^3-42*Tm^2*x+27*Tm*x^2-7*Tm^2+2*Tm*x+27*x^2-5*Tm+34*x+11);

Parametrisation of x (from the Gessel paper)

xTmVm:=(1-Tm+3*Vm+Vm*Tm)/Vm^2/(3+Vm+Tm-Vm*Tm);

KigsKiomSSNUbUc2IiIiIkkjVm1HRiZGJ0YnRiUhIiIqJiIiJEYnRihGJ0YnRidGJ0YnKUYoIiIjRiksKkYkRilGJUYnRihGJ0YrRidGKQ==

Expansion of Vm in terms of (Tm-1)

with(gfun): VmserTm:=proc(n) series(Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Vm=Tm1*VmsTm1,numer(x-xTmVm)), Tm1,VmsTm1,n,true)),Tm1,n+2); end:

n:=10: normal(series(subs(Tm1=Tmser(n)-1,VmserTm(n)),t,1+n));

Ky1JInRHNiIiIiIiIiMsJiIiKUYlSSJ4R0YkRiUiIiUsKComRiZGJSlGKUYmRiVGJSomIiM7RiVGKUYlRiUiIyMpRiUiIicsKiomIiImRiUpRikiIiRGJUYlKiYiI1tGJUYtRiVGJSomIiRGI0YlRilGJUYlIiRXKkYlRigtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

M0xftxTmVm := -(1/2)*(Tm*Vmd-Tm+3*Vmd+1)*(Tm*Vmd^2+2*Tm*Vmd-Vmd^2-Tm-2*Vmd-3)/((Vmd+1)*(Tm^2*Vmd^2-2*Tm^2*Vmd+2*Tm*Vmd^2+Tm^2-3*Vmd^2+2*Tm-6*Vmd-3));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsLiomRilGJSlGK0YmRiVGJSooRiZGJUYpRiVGK0YlRiUqJEYxRiVGLEYpRiwqJkYmRiVGK0YlRixGLkYsRiUsJkYrRiVGJUYlRiwsMiomKUYpRiZGJUYxRiVGJSooRiZGJUY4RiVGK0YlRiwqKEYmRiVGKUYlRjFGJUYlKiRGOEYlRiUqJkYuRiVGMUYlRiwqJkYmRiVGKUYlRiUqJiIiJ0YlRitGJUYsRi5GLEYsRiw=

SsTmVm := -(1/2)*(Tm*Vm-Tm+3*Vm+1)*(Tm*Vm^2+2*Tm*Vm-Vm^2-Tm-2*Vm-3)/((Vm+1)*(Tm^2*Vm^2-2*Tm^2*Vm+2*Tm*Vm^2+Tm^2-3*Vm^2+2*Tm-6*Vm-3));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJI1ZtR0YqRiVGJUYpISIiKiYiIiRGJUYrRiVGJUYlRiVGJSwuKiZGKUYlKUYrRiZGJUYlKihGJkYlRilGJUYrRiVGJSokRjFGJUYsRilGLComRiZGJUYrRiVGLEYuRixGJSwmRitGJUYlRiVGLCwyKiYpRilGJkYlRjFGJUYlKihGJkYlRjhGJUYrRiVGLCooRiZGJUYpRiVGMUYlRiUqJEY4RiVGJSomRi5GJUYxRiVGLComRiZGJUYpRiVGJSomIiInRiVGK0YlRixGLkYsRixGLA==

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

f13s;

LCoqJiwoLUkjU3NHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwKiQsJkYpRipGKkYqRjBGMEYqKUkkRGVzR0YnI0YqRixGKkYqKiYsJiomIiIkRipGKUYqRjBGOUYwRiotSSNSc0dGJ0YoRipGKiomRjlGKkkjUjBHRidGKkYwRjFGKg==

We need to express R0

R0sol;

LCQqJiMiIiIiIiRGJS1JI1NzRzYiNiMhIiJGJUYr

factor(subs(x=-1,Ssx=Ss(-1),algSTm));

LDAqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKS1JI1NzR0YoNiMhIiIiIiNGJUYuKihGJEYlRiZGJUYrRiVGLiomIiIlRiUpRidGMkYlRiUqJiIjW0YlRiZGJUYuKiYiI0NGJSlGJ0YvRiVGJSomIiM7RiVGJ0YlRiVGMkYl

factor(subs(Tm=Zm^2,%));

LCQqKCIiJSIiIiwsKigiIilGJSlJI1ptRzYiIiIkRiUtSSNTc0dGKzYjISIiRiVGJSokKUYqRiRGJUYlKiZGJEYlRilGJUYlKiYiIiNGJSlGKkY1RiVGJUYlRiVGJSwsRidGJUYxRjBGM0YlRjRGMEYlRjBGJUYw

Ssm1solens:=factor([solve(%,Ss(-1))]);

NyQsJCoqIyIiIiIiKUYmLCZJI1ptRzYiRiZGJiEiIkYmLCoqJClGKSIiJEYmRiYqJkYvRiYpRikiIiNGJkYrRilGK0YmRitGJkYuRitGJiwkKipGJUYmLCZGKUYmRiZGJkYmLCpGLUYmRjBGJkYpRitGJkYmRiZGLkYrRis=

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

Zmser(10);

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

subs(x=-1,Ssser(5));

LCgqJiIjSiIiIilJInRHNiIiIidGJSEiIiomIiIlRiUpRidGLEYlRioqJClGJyIiI0YlRio=

n:=10: normal(series(subs(Zm=Zmser(n), Ssm1solens[1]),t,n+1)); normal(series(subs(Zm=Zmser(n), Ssm1solens[2]),t,n+1));

Ky1JInRHNiIhIiIiIiMhIiUiIiUhI0oiIichJCwkIiIpLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNQ==

Ky9JInRHNiIhIiIiIiEiIiIiIiMiIiVGKSIjSiIiJyIkLCQiIiktSSJPRyUqcHJvdGVjdGVkRzYjRiciIzU=

Ssm1Zm:=Ssm1solens[1];

LCQqKiMiIiIiIilGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKkYoRipGJUYqRiVGLUYqRiU=

R0Zm:=-Ssm1Zm/3;

LCQqKiMiIiIiI0NGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKkYoRipGJUYqRiVGLUYqRio=

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

indets(algSTm);

PCVJJFNzeEc2IkkjVG1HRiRJInhHRiQ=

res:=numer(factor(resultant(subs(Ssx=DD+2*Ss1x+1/(1+x),algSTm),subs(Ssx=Ss1x,x=1/x,algSTm),Ss1x))): nops(res);

IiIl

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

NiYiIiIiIiMiI3ciJEom

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

NiYiIiFGIyIiJCIiJw==

n:=1: normal(series(subs(DD=Ssser(n)-2*subs(x=1/x,Ssser(n))-1/(1+x),Tm=Tmser(n),op(3,res)),t,n+1));

KydJInRHNiIsJCooIiQ3JiIiIkkieEdGJEYoLCgqJClGKSIiJEYoRigqJkYtRihGKUYoISIiIiIjRi9GKEYoIiIhLUkiT0clKnByb3RlY3RlZEc2I0YoRig=

n:=10: normal(series(subs(DD=Ssser(n)-2*subs(x=1/x,Ssser(n))-1/(1+x),Tm=Tmser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

indets(op(4,res));

PCVJI0RERzYiSSNUbUdGJEkieEdGJA==

algDDTm:=op(4,res):

Hence DD has degree 6 over Q(x,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);

IiUwWQ==

degree(res,DD);

IiND

Back to Rs

f13s;

LCoqJiwoLUkjU3NHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwKiQsJkYpRipGKkYqRjBGMEYqKUkkRGVzR0YnI0YqRixGKkYqKiYsJiomIiIkRipGKUYqRjBGOUYwRiotSSNSc0dGJ0YoRipGKiomRjlGKkkjUjBHRidGKkYwRjFGKg==

res:=numer(factor(subs(DD=(3*(x+1)*Rsx+3*R0Zm-1/(1+x))/sqrt(Deltas),t=sqrt(t2Tm),Tm=Zm^2,algDDTm))): nops(%);

IiIm

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

NiciIiJGIyIiIyIkOyIiIyUq

Nic8IjwjSSJ4RzYiRiQ8JUkkUnN4R0YmSSNabUdGJkYlRic=

NiciIiFGI0YjIiIkRiQ=

n:=10: normal(series(subs(Zm=Zmser(n),Rsx=Rsser(n),op(4,res)),t,n+1));

Ky9JInRHNiIsJComIiUnNCUiIiIpSSJ4R0YkIiIjRihGKCIiISwsKiZGJ0YoKUYqIiIlRighIiIqJiImJVE7RigpRioiIiRGKEYxKiYiJmc0JUYoRilGKEYoKiZGM0YoRipGKEYxRidGMUYrLCwqJiImT2InRihGL0YoRjEqJiInV0BFRihGNEYoRjEqJiInM2lnRihGKUYoRigqJkY9RihGKkYoRjFGO0YxRjAsLComIidDJSoqKkYoRi9GKEYxKiYiKCdwKCpSRihGNEYoRjEqJiIoLyY0IipGKEYpRihGKComRkVGKEYqRihGMUZDRjEiIicsLComIilbZzU6RihGL0YoRjEqJiIpIz5DLydGKEY0RihGMSomIiohb1Z2OEYoRilGKEYoKiZGTkYoRipGKEYxRkxGMSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjNQ==

normal(series(subs(Zm=Zmser(n),Rsx=Rsser(n),op(5,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Equation of Rs(x) over Q(x,Z)

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

PCVJJFJzeEc2IkkjWm1HRiRJInhHRiQ=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Equation for Rs(x) over Q(x,t)

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

IiUmNCU=

degree(res,Rsx);

IiND

algRs:=res:

n:=10:normal(series(subs(Rsx=Rsser(n),algRs),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

Parametrization by V

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

IiIk

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

NiUiIiIiI1kiJFok

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

NiU8IjwlSSRSc3hHNiJJI1ZtR0YmSSNabUdGJkYk

NiUiIiEiIiIiIiM=

n:=10: normal(series(subs(Vm=VmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rsx=Rsser(n),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=22: normal(series(subs(Vm=VmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rsx=Rsser(n),op(3,res)),t,n+1));

KylJInRHNiIsKComIiZvRiQiIiIpSSJ4R0YkIiIkRighIiIqJkYnRigpRioiIiNGKEYsKiYiJk9iJ0YoRipGKEYoIiM/LCwqJiInN1xIRigpRioiIiVGKEYsKiYiKCtvRiRGKEYpRihGLComIigrd1gjRihGLkYoRiwqJiIoV2wqZkYoRipGKEYoRidGKCIjQS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjQw==

Rational expression of Rs in terms of Zm and Vm

RsZmVm:=factor(solve(op(2,res),Rsx));

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

Back to M(x,0)

op(Rsser); op(Rser);

Zio2I0kibkc2IkYlRiVGJS1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ4R0YlLUklc3FydEc2JEYoSShfc3lzbGliR0YlNiNGKy1JJVJzZXJHRiVGI0YlRiVGJQ==

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSooKUkidEdGJSIiIyIiIkkieEdGJSEiIi1JJXN1YnNHRik2JC9JInlHRiUiIiEtSSVNc2VyR0YlNiMsJkYkRjBGMEYyRjBGLkknZmFjdG9yR0YoRiVGJUYl

Expression of t^2M(x,0)/x (Vmd is Vm evaluated at x^2 rather than x)

Mx0sxft2ZmVm:=factor(subs(Vm=Vmd,RsZmVm));

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

Let us check

n:=12: normal(series(subs(Vmd=subs(x=x^2,VmserTm(n)),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Mx0sxft2ZmVm),t,n+1));

Ky9JInRHNiIjIiIiIiIkIiIjLCYqJkYlRiYpSSJ4R0YkRihGJkYmIyIiJUYnRiZGLiwoKiYjRihGJ0YmKUYsRi5GJkYmKiZGLkYmRitGJkYmIyIjSkYnRiYiIicsKiomIyIiJkYnRiYpRixGNkYmRiYqJiMiI1NGJ0YmRjJGJkYmKiYiI1pGJkYrRiZGJiMiJCwkRidGJiIiKSwsKiYjIiM5RidGJilGLEZDRiZGJiomIyIkUyJGJ0YmRjtGJkYmKiYjIiRKJ0YnRiZGMkYmRiYqJiMiJTo8RidGJkYrRiZGJiMiJSdHJEYnRiYiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIzc=

subs(y=0,t^2*Mser(8)/x);

KigpSSJ0RzYiIiIjIiIiLCwqKiNGJyIiJEYnSSJ4R0YlRicsLComIiM5RicpRiwiIilGJ0YnKiYiJFMiRicpRiwiIidGJ0YnKiYiJEonRicpRiwiIiVGJ0YnKiYiJTo8RicpRixGJkYnRiciJSdHJEYnRicpRiRGMUYnRicqKkYqRidGLEYnLCoqJiIiJkYnRjRGJ0YnKiYiI1NGJ0Y4RidGJyomIiRUIkYnRjxGJ0YnIiQsJEYnRicpRiRGNUYnRicqKkYqRidGLEYnLCgqJkYmRidGOEYnRicqJiIjN0YnRjxGJ0YnIiNKRidGJylGJEY5RidGJyomLCYqJkYqRicpRixGK0YnRicqJiNGOUYrRidGLEYnRidGJ0YjRidGJyomRipGJ0YsRidGJ0YnRiwhIiI=

normal(series(%%-%,t,9));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

factor(subs(Tm=Zm^2,t2Tm));

LCQqLCMiIiIiJGMjRiUsJkkjWm1HNiJGJUYlISIiRiUsJkYoRiVGJUYlRiUpLCYqJClGKCIiI0YlRiUiIiRGJUYxRiUpRigiIidGKkYl

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

Zmser(10);

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

R0Zm:=-(1/24)*(Zm-1)*(Zm^3-3*Zm^2-Zm-1)/Zm^3;

LCQqKiMiIiIiI0NGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKkYoRipGJUYqRiVGLUYqRio=

algRsZm:=55296*Zm^9*x*(x+1)^4*Rsx^3-6912*Zm^6*x*(x+1)^2*(Zm^4*x+Zm^4-4*Zm^3*x+4*Zm^3+2*Zm^2*x+2*Zm^2+x+1)*Rsx^2+24*Zm^3*(Zm^8*x^4-2*Zm^8*x^2-96*Zm^7*x^3+8*Zm^6*x^4+224*Zm^6*x^3+Zm^8+96*Zm^7*x-80*Zm^6*x^2-192*Zm^5*x^3+18*Zm^4*x^4+224*Zm^6*x+96*Zm^4*x^3+8*Zm^6+192*Zm^5*x+156*Zm^4*x^2-96*Zm^3*x^3+96*Zm^4*x-96*Zm^2*x^3+18*Zm^4+96*Zm^3*x-192*Zm^2*x^2-27*x^4-96*Zm^2*x-96*x^3-138*x^2-96*x-27)*Rsx+(Zm-1)*(Zm+1)*(Zm^10*x^3+15*Zm^10*x^2+4*Zm^9*x^3+15*Zm^10*x-12*Zm^9*x^2+11*Zm^8*x^3+Zm^10+12*Zm^9*x-91*Zm^8*x^2+36*Zm^7*x^3-4*Zm^9-91*Zm^8*x+276*Zm^7*x^2+46*Zm^6*x^3+11*Zm^8-276*Zm^7*x-270*Zm^6*x^2+108*Zm^5*x^3-36*Zm^7-270*Zm^6*x+444*Zm^5*x^2+90*Zm^4*x^3+46*Zm^6-444*Zm^5*x-122*Zm^4*x^2+108*Zm^3*x^3-108*Zm^5-122*Zm^4*x+60*Zm^3*x^2+81*Zm^2*x^3+90*Zm^4-60*Zm^3*x+127*Zm^2*x^2-108*Zm^3+127*Zm^2*x+27*x^3+81*Zm^2+85*x^2+85*x+27);

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

RsZmVm := -(1/48)*Vm*(Vm*Zm^2-Zm^2-Vm-3)*(Vm^3*Zm^4-2*Vm^3*Zm^3-3*Vm^2*Zm^4+4*Vm^3*Zm^2-6*Vm^2*Zm^3-3*Vm*Zm^4-6*Vm^3*Zm+8*Vm^2*Zm^2+6*Vm*Zm^3+Zm^4+3*Vm^3-10*Vm^2*Zm+8*Vm*Zm^2+2*Zm^3+11*Vm^2+10*Vm*Zm+4*Zm^2+11*Vm+6*Zm+3)/(Zm^3*(Vm+1)^2*(Vm*Zm-Vm-Zm-1)^2);

Mx0sxft2ZmVm:=-(1/48)*Vmd*(Vmd*Zm^2-Zm^2-Vmd-3)*(Vmd^3*Zm^4-2*Vmd^3*Zm^3-3*Vmd^2*Zm^4+4*Vmd^3*Zm^2-6*Vmd^2*Zm^3-3*Vmd*Zm^4-6*Vmd^3*Zm+8*Vmd^2*Zm^2+6*Vmd*Zm^3+Zm^4+3*Vmd^3-10*Vmd^2*Zm+8*Vmd*Zm^2+2*Zm^3+11*Vmd^2+10*Vmd*Zm+4*Zm^2+11*Vmd+6*Zm+3)/(Zm^3*(Vmd+1)^2*(Vmd*Zm-Vmd-Zm-1)^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;

expression of E:=2K(x,y)M(x,y)

E1:=collect(-subs(M(x,y)=0,eqcatM),[M,M10],factor);;

LCwqKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnRi1GLkYrRi4tSSJNR0YpNiQiIiFGK0YnRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJy1GMTYkRi1GM0YnRi1GLkYrRi5GLioqRihGJywmKiRGKkYnRidGJ0YnRictRjE2JEYrRjNGJ0YqRi5GJyomI0YmIiIkRidGLUYnRicqKEYoRidGK0YuSSRNMTBHRilGJ0Yu

We will determine the degree of E over Q(t) when x=2 and y=3

E2:=subs(x=2,y=3,E1);

LCwqKCMiIiIiIidGJSwmKiYiIz9GJUkidEc2IkYlRiVGJiEiIkYlLUkiTUdGKzYkIiIhIiIkRiVGLCooIyIiJkYxRiVGKkYlLUYuNiQiIiNGMEYlRiwqKCMiIzUiIipGJUYqRiUtRi42JEYxRjBGJUYlIyIiJUYxRiUqKCNGJUYxRiVGKkYlSSRNMTBHRitGJUYs

Introduce the series Rs and Ss

E3:=subs(M10=R0/t^2,M(2,0)=Rs(4)/t^2*2,M(3,0)=Rs(9)/t^2*3,M(0,3)=Ss(9)/t/3,%);

LCwqKiMiIiIiIz1GJSwmKiYiIz9GJUkidEc2IkYlRiUiIichIiJGJS1JI1NzR0YrNiMiIipGJUYqRi1GLSooIyIjNSIiJEYlRipGLS1JI1JzR0YrNiMiIiVGJUYtKihGM0YlRipGLS1GN0YwRiVGJSNGOUY1RiUqKCNGJUY1RiVGKkYtSSNSMEdGK0YlRi0=

We have an expression of R0 in terms of Zm, and then cubic equations for Ss (resp Rs) over Q(x,T) (resp. Q(x,Z))

<Text-field style="Heading 3" layout="Heading 3">Successive elimination of the series Ss(9), Rs(4), Rs(9)</Text-field>

algSsTm;collect(algRsZm,Rsx,factor);

res1:=factor(resultant(subs(Ss(9)=Ssx,R0=R0Zm,E-E3),subs(x=9,Tm=Zm^2,algSsTm),Ssx)): nops(res1);

IiIl

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

NiYiIiMiJCNHRiNGIw==

NiY8IjwlSSJFRzYiSSNabUdGJkkidEdGJjwjRic8I0Yo

res2:=op(2,res1):

res3:=factor(resultant(subs(Rs(9)=Rsx,res2),subs(x=9,algRsZm),Rsx)): nops(res3);

IiIl

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiYiIiIiIiMiJHoiIiVHOw==

NiY8IjwjSSNabUc2IjwlSSJFR0YmRiVJInRHRiZGJw==

Choosing the correct factor

n:=10: normal(series(subs(Zm=Zmser(n),E=2*K*Mser(n),x=2,y=3,Rs(4)=subs(x=4,Rsser(n)),op(3,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=5:normal(series(subs(Zm=Zmser(n),E=2*K*Mser(n),x=2,y=3,Rs(4)=subs(x=4,Rsser(n)),op(4,res3)),t,n+1));

Ky9JInRHNiIhMDN1dE1mRTopIiIhIjI3QD8hcHhCIz0iIiIiITJTJzNpIlxaKWUhKiIiIyIzd2xIZXJbUGtaIiIkITRvNnohKipwTChHayIiIiUtSSJPRyUqcHJvdGVjdGVkRzYjRigiIiY=

res4:=op(3,res3):

res5:=factor(resultant(subs(Rs(4)=Rsx,res3),subs(x=4,algRsZm),Rsx)): nops(res5);

IiIl

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiYiIiIiIiMiJUE9IiZmSiI=

NiY8IjwjSSNabUc2IjwlSSJFR0YmRiVJInRHRiZGJw==

n:=10: normal(series(subs(Zm=Zmser(n),E=2*K*Mser(n),x=2,y=3,op(3,res5)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=5:normal(series(subs(Zm=Zmser(n),E=2*K*Mser(n),x=2,y=3,op(4,res5)),t,n+1));

Ky9JInRHNiIhWG9KQWBULUByKEhfUiI9a0coWz0mUjEiUTZISHZwOSIiISJZK28qKkgmeSgpMzY8PXUleTBSaXNvS0R5eHQhM0E/ZygiIiIhZW53cClRYFMiNHkjb3ksJWYnM3hcNGhSLnRnN3RmYzkmPSIiIyJmbl82X2Bwb3EmZmVaSW5LTyRSNyFwKm9dalByPG8lUllHIiIkIWduP143WGlQIz5RJW8wQiRwSykqSCRmRyZRMSE0WEdlQEZMSiIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIiJg==

res6:=op(3,res5):

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

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

res7:=factor(resultant(res6,algZm,Zm)): nops(res7);

IiIk

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NiUiIiJGIyIlIT0n

NiU8IjwjSSJ0RzYiPCRJIkVHRiZGJQ==

degree(op(3,res7),E);

IiNz

algE:=op(3,res7):

n:=20: normal(series(subs(E=2*K*Mser(n),x=2,y=3,algE),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

<Text-field style="Heading 3" layout="Heading 3">Similar procedure for P(x,y), with 4 series to eliminate</Text-field>

Now for P(x,y)

PexprM;

LCoqJkkieEc2IiEiIi1JIk1HRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1GKDYkRipGJEYrRisqJkYqRiYtRig2JEYvRiRGK0Ym

Expression of P(x,y) in terms of the specializations of M

eqcatP:=collect(subs(isolate(PexprM-P,M(x,y)),eqcatM/x+subs({x=y,y=x},eqcatM/x)),[P,M,M10],factor);

LDAqLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiVGK0YxRi1GMUkiUEdGKUYlRjEqKkYtRjEsKCoqRiRGJUYoRiVGKkYlRi1GJUYlKihGJEYlRihGJUYtRiVGJUYrRjFGJUYrRjEtSSJNR0YpNiQiIiFGK0YlRjEqKiwoKipGJEYlRihGJUYrRiVGLEYlRiUqKEYkRiVGKEYlRitGJUYlRi1GMUYlRi1GMUYrRjEtRjg2JEY6Ri1GJUYxKixGKEYlLCYqJEYqRiVGJUYlRiVGJUYqRjFGLUYxLUY4NiRGK0Y6RiVGJSosRihGJSwmKiRGLEYlRiVGJUYlRiVGK0YxRixGMS1GODYkRi1GOkYlRiUjIiIlIiIkRjEqLEYkRiVGKEYlRitGMUYtRjFJJE0xMEdGKUYlRiU=

expression of F:=2K(x,y)P(x,y)

F1:=collect(-subs(P=0,eqcatP),[M,M10],factor);;

LC4qKkkieUc2IiEiIiwoKioiIiMiIiJJInRHRiVGKilJInhHRiVGKUYqRiRGKkYqKihGKUYqRitGKkYkRipGKkYtRiZGKkYtRiYtSSJNR0YlNiQiIiFGLUYqRioqKiwoKipGKUYqRitGKkYtRiopRiRGKUYqRioqKEYpRipGK0YqRi1GKkYqRiRGJkYqRiRGJkYtRiYtRjA2JEYyRiRGKkYqKixGK0YqLCYqJEYsRipGKkYqRipGKkYsRiZGJEYmLUYwNiRGLUYyRipGJiosRitGKiwmKiRGNkYqRipGKkYqRipGLUYmRjZGJi1GMDYkRiRGMkYqRiYjIiIlIiIkRioqLEYpRipGK0YqRi1GJkYkRiZJJE0xMEdGJUYqRiY=

We will determine the degree of F over Q(t) when x=2 and y=3

F2:=subs(x=2,y=3,F1);

LC4qKCMiIiIiIidGJSwmKiYiI0lGJUkidEc2IkYlRiUiIiMhIiJGJS1JIk1HRis2JCIiIUYsRiVGJSooRiRGJSwmKiYiI1NGJUYqRiVGJSIiJEYtRiUtRi82JEYxRjZGJUYlKigjIiImIiM3RiVGKkYlLUYvNiRGLEYxRiVGLSooI0Y7IiIqRiVGKkYlLUYvNiRGNkYxRiVGLSMiIiVGNkYlKigjRiVGNkYlRipGJUkkTTEwR0YrRiVGLQ==

F3:=subs(M10=R0/t^2,M(2,0)=Rs(4)/t^2*2,M(3,0)=Rs(9)/t^2*3,M(0,3)=Ss(9)/t/3,M(0,2)=Ss(4)/t/2,F2);

LC4qKiMiIiIiIzdGJSwmKiYiI0lGJUkidEc2IkYlRiUiIiMhIiJGJS1JI1NzR0YrNiMiIiVGJUYqRi1GJSoqI0YlIiM9RiUsJiomIiNTRiVGKkYlRiUiIiRGLUYlLUYvNiMiIipGJUYqRi1GJSooIyIiJiIiJ0YlRipGLS1JI1JzR0YrRjBGJUYtKigjRj5GOEYlRipGLS1GQUY6RiVGLSNGMUY4RiUqKCNGJUY4RiVGKkYtSSNSMEdGK0YlRi0=

res1:=factor(resultant(subs(Ss(9)=Ssx,R0=R0Zm,F-F3),subs(x=9,Tm=Zm^2,algSsTm),Ssx)): nops(res1);

IiIl

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

NiYiIiMiJFkmRiNGIw==

NiY8IjwlSSJGRzYiSSNabUdGJkkidEdGJjwjRic8I0Yo

res2:=op(2,res1):

res3:=factor(resultant(subs(Rs(9)=Rsx,res2),subs(x=9,algRsZm),Rsx)): nops(res3);

IiIl

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiYiIiIiIiMiJCNRIiVaZA==

NiY8IjwjSSNabUc2IjwlSSJGR0YmRiVJInRHRiZGJw==

Choosing the correct factor

n:=10: normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),x=2,y=3,Rs(4)=subs(x=4,Rsser(n)),Ss(4)=subs(x=4,Ssser(n)),op(3,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=5:normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),x=2,y=3,Rs(4)=subs(x=4,Rsser(n)),Ss(4)=subs(x=4,Ssser(n)),op(4,res3)),t,n+1));

Ky9JInRHNiIhLykzQDQ3YTQmIiIhIjFHMGJBV2ZiSCIiIiEyU1U6WSVcJz1XJyIiIyIzV0BJJD1TdjNoJyIiJCE0WytgJVxgO3VaTSIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIiJg==

res4:=op(3,res3):

res5:=factor(resultant(subs(Rs(4)=Rsx,res3),subs(x=4,algRsZm),Rsx)): nops(res5);

IiIl

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiYiIiIiIiMiJScqKioiJyIqUjk=

NiY8IjwjSSNabUc2IjwlSSJGR0YmRiVJInRHRiZGJw==

n:=10: normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),Ss(4)=subs(x=4,Ssser(n)),x=2,y=3,op(3,res5)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=1:normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),Ss(4)=subs(x=4,Ssser(n)),x=2,y=3,op(4,res5)),t,n+1));

KydJInRHNiIiWFN1MiFSQTg9KCl5ViYqcCt3Z2A7IUgnUkZlMSYpW2w8IiIhLUkiT0clKnByb3RlY3RlZEc2IyIiIkYr

res6:=op(3,res5):

res7:=factor(resultant(subs(Ss(4)=Ssx,res6),subs(x=4,Tm=Zm^2,algSsTm),Ssx)): nops(res7);

IiIk

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NiUiIiIiJWBIIiYjb0A=

NiU8IjwlSSJGRzYiSSNabUdGJkkidEdGJkYk

n:=10: normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),x=2,y=3,op(2,res7)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=1: normal(series(subs(Zm=Zmser(n),F=2*K*Pser(n),x=2,y=3,op(3,res7)),t,n+1));

KydJInRHNiIhZm8pM1hCYiRcMTlyRUsmUkxLdW1OMDU/RVsqUihlbDVdKTMrYVlwZDUrIiIiIS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiJGKw==

res8:=op(2,res7):

Final step

res9:=factor(resultant(res8,algZm,Zm)): nops(res9);

IiIk

seq(nops(op(i,res9)),i=1..nops(res9)); seq(indets(op(i,res9)),i=1..nops(res9));

NiUiIiJGIyIlZycq

NiU8IjwjSSJ0RzYiPCRJIkZHRiZGJQ==

degree(op(3,res9),F);

IiNz

algF:=op(3,res9):

n:=20: normal(series(subs(F=2*K*Pser(n),x=2,y=3,algF),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

This can't be optimal... For instance Ss(9) and Rs(9) belong to a common extension of Q(x,t) of degree24

somme:=op(1,E3)+op(3,E3);

LCYqKiMiIiIiIz1GJSwmKiYiIz9GJUkidEc2IkYlRiUiIichIiJGJS1JI1NzR0YrNiMiIipGJUYqRi1GLSooIyIjNSIiJEYlRipGLS1JI1JzR0YrRjBGJUYl

res1:=factor(resultant(subs(Ss(9)=Ssx,som-somme),subs(x=9,algSsTm),Ssx)): nops(res1);

IiIk

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

NiUiIiMiIyEqRiM=

NiU8IjwlSSNUbUc2Ikkkc29tR0YmSSJ0R0YmPCNGKA==

res2:=op(2,res1):

res3:=factor(resultant(subs(Rs(9)=Rsx,Tm=Zm^2,res2),subs(x=9,algRsZm),Rsx)): nops(res3);

IiIk

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiUiIiIiJDAiIiQ/Jw==

NiU8IjwlSSNabUc2Ikkkc29tR0YmSSJ0R0YmRiQ=

n:=10: series(subs(Zm=Zmser(n), som=somme,Ss(9)=subs(x=9,Ssser(n)), Rs(9)=subs(x=9,Rsser(n)), op(2,res3)),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

series(subs(Zm=Zmser(n), som=somme,Ss(9)=subs(x=9,Ssser(n)), Rs(9)=subs(x=9,Rsser(n)), op(3,res3)),t,n+1);

KzlJInRHNiIhMDN1dE1mRTopIiIhIjI3QD8hcHhCIz0iIiIiITJTJzNpIlxaKWUhKiIiIyIzd2xIZXJbUGtaIiIkITRvNnohKipwTChHayIiIiUiNGsvJG9QdlIhPSg9IiImIjUnSE5wWSp5MCIpUk4iIichNiU9KD5mJFInZSJRTFIiIigiNztfQUwpKlxfMzA3SiIiKSE4V0AlPk5wQmsqKVEuPyIiKi1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjNQ==

res4:=op(2,res3):nops(%);

IiQwIg==

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

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

algsom:=res4: degree(algsom,som);

IiIk

E3;

LCwqKiMiIiIiIz1GJSwmKiYiIz9GJUkidEc2IkYlRiUiIichIiJGJS1JI1NzR0YrNiMiIipGJUYqRi1GLSooIyIjNSIiJEYlRipGLS1JI1JzR0YrNiMiIiVGJUYtKihGM0YlRipGLS1GN0YwRiVGJSNGOUY1RiUqKCNGJUY1RiVGKkYtSSNSMEdGK0YlRi0=

E4:=subs(R0=R0Zm,som+subs(Ss(9)=0,Rs(9)=0,E3));

LCpJJHNvbUc2IiIiIiMiIiUiIiRGJSooIyIjNUYoRiVJInRHRiQhIiItSSNSc0dGJDYjRidGJUYtKiwjRiUiI3NGJUYsRi0sJkkjWm1HRiRGJUYlRi1GJSwqKiQpRjVGKEYlRiUqJkYoRiUpRjUiIiNGJUYtRjVGLUYlRi1GJUY4Ri1GJQ==

res5:=factor(resultant(subs(Rs(4)=Rsx,E-E4),subs(x=4,algRsZm),Rsx)): nops(res5);

IiIk

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiUiIiMiI3FGIw==

NiU8IjwmSSJFRzYiSSNabUdGJkkkc29tR0YmSSJ0R0YmPCNGKQ==

res6:=op(2,res5):

res7:=factor(resultant(res6,algsom,som)): nops(res7);

IiIl

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NiYiIiIiIiNGJCIlQT0=

NiY8IjwjSSNabUc2IjwjSSJ0R0YmPCVJIkVHRiZGJUYo

res8:=op(4,res7):

res9:=factor(resultant(res8,algZm,Zm)): nops(res9);

IiIk

seq(nops(op(i,res9)),i=1..nops(res9)); seq(indets(op(i,res9)),i=1..nops(res9));

NiUiIiJGIyIlIT0n

NiU8IjwjSSJ0RzYiPCRJIkVHRiZGJQ==

algE:=op(3,res9):

degree(algE,E);

IiNz

n:=20:normal(series(subs(E=2*K*Mser(n),x=2,y=3,algE),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

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

We start with Rs(x) and the algebraic equation (degree 3) it satisfies over Q(Zm,x)

res:=factor(resultant(numer(Rsx-RsZmVm),subs(Tm=Zm^2,numer(x-xTmVm)),Vm)):nops(%);

IiIm

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

NiciIiIiIiNGJEYkIiMlKg==

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

NiYhI0sqJCksJiokKUkjWm1HNiIiIiMiIiJGLEYsRixGK0YsKiQpLCZGKUYsRiwhIiIiIiZGLCokKSwmRilGLEYsRixGMUYs

algRsZm:=op(5,res): degree(%,Rsx);

IiIk

Degree 1 when x=0

collect(factor(subs(x=0,algRsZm)),Rsx,factor);

LCYqLiIjQyIiIiwmSSNabUc2IkYlRiUhIiJGJSwmRidGJUYlRiVGJSksJiokKUYnIiIjRiVGJSIiJEYlRjBGJSlGJ0YwRiVJJFJzeEdGKEYlRiUqKilGJkYvRiVGKkYlRitGJSwqKiRGMUYlRiUqJkYwRiVGLkYlRilGJ0YpRiVGKUYlRiU=

degree(%,Rsx);

IiIi

factor(solve(%%,Rsx));

LCQqKiMiIiIiI0NGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKkYoRipGJUYqRiVGLUYqRio=

Now for Ss(x) (recall it is a multiple of x), we find that the coefficients are rational functions of T

Ssser(4);

LCYqKEkieEc2IiIiIiwmKiYiIiRGJkYkRiZGJiIiKEYmRiYpSSJ0R0YlIiIlRiZGJiomKUYsIiIjRiZGJEYmRiY=

res:=factor(resultant(numer(Ssx-SsTmVm),numer(x-xTmVm),Vm)):nops(%);

IiIl

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

NiYiIiIiIiNGJCIjaw==

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

NiUhIzsqJCksJkkjVG1HNiIiIiJGKSEiIiIiI0YpKiQpLCZGJ0YpRilGKUYrRik=

algSsVm:=op(4,res): degree(%,Ssx);

IiIk

An equation for Ss(x)/x

algSssxVm:=collect(op(2,factor(subs(Ssx=x*Sssx,algSsVm))),Sssx,factor);

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

collect(factor(subs(x=0,algSssxVm)),Sssx,factor);

LCYqKCwmSSNUbUc2IiIiIkYnISIiRicpLCZGJUYnIiIkRidGK0YnSSVTc3N4R0YmRidGJyomKUYkIiIjRicsKCokKUYlRi9GJ0YnKiYiIidGJ0YlRidGKCIjNkYoRidGJw==

degree(%,Sssx);

IiIi

<Text-field style="Heading 3" layout="Heading 3">Walks ending at abscissa -1: the series M_{0,i}, i odd</Text-field>

Sssxser:=map(factor,op(algeqtoseries(algSssxVm,x,Sssx,2,true)));

KylJInhHNiIsJCooLCZJI1RtR0YkIiIiRikhIiJGKSwoKiQpRigiIiNGKUYpKiYiIidGKUYoRilGKiIjNkYqRikpLCZGKEYpIiIkRilGNEYqRioiIiEqKCwsKiQpRigiIiVGKUYpKiYiI1dGKSlGKEY0RilGKiomIiQ1IkYpRi1GKUYpKiYiJD8lRilGKEYpRikiJCJHRilGKSlGJ0YuRikpRjNGMEYqRiktSSJPRyUqcHJvdGVjdGVkRzYjRilGLg==

S1Tm:=coeff(Sssxser,x,0); S3Tm:=coeff(Sssxser,x,1);

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

KigsLCokKUkjVG1HNiIiIiUiIiJGKSomIiNXRikpRiYiIiRGKSEiIiomIiQ1IkYpKUYmIiIjRilGKSomIiQ/JUYpRiZGKUYpIiQiR0YpRikpLCZGJkYpRilGLkYyRikpLCZGJkYpRi1GKSIiJ0Yu

M01ftTm:=S1Tm: M03ftTm:=S3Tm:

Walks ending at (-1,1) or (-1,3)

Cm11ftsol:=M01ftTm: Cm13ftsol:=M03ftTm:

n:=10: normal(series(subs(Tm=Tmser(n),Cm11ftsol),t,n+1));

Ky1JInRHNiIiIiIiIiMiIigiIiUiI2wiIiciJCNwIiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

n:=10: coeff(coeff(convert(Cser(n),polynom),x,-1),y,1);

LCwqJiIlIip6IiIiKUkidEc2IiIiKkYlRiUqJiIkI3BGJSlGJyIiKEYlRiUqJiIjbEYlKUYnIiImRiVGJSomRi1GJSlGJyIiJEYlRiVGJ0Yl

n:=10: normal(series(subs(Tm=Tmser(n),Cm13ftsol),t,n+1));

Ky1JInRHNiIiIiQiIiUiI1ciIiciJCJlIiIpIiVHdyIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

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

LCoqJiIlR3ciIiIpSSJ0RzYiIiIqRiVGJSomIiQiZUYlKUYnIiIoRiVGJSomIiNXRiUpRiciIiZGJUYlKiYiIiRGJSlGJ0YzRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0), (-4,0), (-6,0), (-8,0)</Text-field>

ser:=map(factor,op(algeqtoseries(algRsZm,x,Rsx,4,true)));

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

M10Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,0)/t2Tm)); M30Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,1)/t2Tm)); M50Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,2)/t2Tm)); M70Zm:=factor(subs(Tm=Zm^2,coeff(ser,x,3)/t2Tm));

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYw

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNBRicpRikiIidGJ0YnKiYiI2FGJylGKSIiJkYnRiwqJiIkMyJGJylGKSIiJUYnRicqJkY6RidGKEYnRiwqJiIkSSJGJylGKSIiI0YnRicqJkYyRidGKUYnRixGPEYsRicsJkYpRidGJ0YnRiwpLCYqJEZERidGJ0YmRidGOEYsRic=

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicpLCZGKUYnRichIiIiIiNGJyw+KiQpRikiIzhGJ0YnKiYiIzxGJylGKSIjN0YnRi0qJiIjaUYnKUYpIiM2RidGJyomIiRBIkYnKUYpIiM1RidGLSomIiRKJkYnKUYpIiIqRidGJyomIiR6J0YnKUYpIiIpRidGLSomIiVDOkYnKUYpIiIoRidGJyomIiVnT0YnKUYpIiInRidGLSomIiU+PEYnKUYpIiImRidGJyomIiVSbUYnKUYpIiIlRidGLSomIiVFOEYnRihGJ0YnKiYiJVlORicpRilGLkYnRi0qJiIkIikqRidGKUYnRiciJEYkRidGJywmRilGJ0YnRidGLSksJiokRmVuRidGJ0YmRidGQkYtRi0=

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicpLCZGKUYnRichIiJGJkYnLEgqJClGKSIjPUYnRicqJiIjQ0YnKUYpIiM8RidGLSomIiQ+IkYnKUYpIiM7RidGJyomIiRLIkYnKUYpIiM6RidGLSomIiUpMyJGJylGKSIjOUYnRicqJiIlR01GJylGKSIjOEYnRi0qJiIlN0pGJylGKSIjN0YnRicqJiImL2cjRicpRikiIzZGJ0YtKiYiJnkkPUYnKUYpIiM1RidGJyomIiZjLiVGJylGKSIiKkYnRi0qJiInVTA4RicpRikiIilGJ0YnKiYiJTtsRicpRikiIihGJ0YnKiYiJ1dKSUYnKUYpIiInRidGJyomIiYnUkBGJylGKSIiJkYnRicqJiInczFGRicpRikiIiVGJ0YnKiYiJkNUJUYnRihGJ0YtKiYiJjAxKEYnKUYpIiIjRidGJyomIiY7XCVGJ0YpRidGLSImSDciRi1GJywmRilGJ0YnRidGLSksJiokRmhvRidGJ0YmRidGSUYtRic=

Cm20sol:=M10Zm-Q00/3; Cm40sol:=M30Zm-Q20/3; Cm60sol:=M50Zm-Q40/3; Cm80sol:=M70Zm-Q60/3;

LCYqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYwKiYjRidGJkYnSSRRMDBHRipGJ0Yw

LCYqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNBRicpRikiIidGJ0YnKiYiI2FGJylGKSIiJkYnRiwqJiIkMyJGJylGKSIiJUYnRicqJkY6RidGKEYnRiwqJiIkSSJGJylGKSIiI0YnRicqJkYyRidGKUYnRixGPEYsRicsJkYpRidGJ0YnRiwpLCYqJEZERidGJ0YmRidGOEYsRicqJiNGJ0YmRidJJFEyMEdGKkYnRiw=

LCYqLiMiI0siIiQiIiIpSSNabUc2IkYmRicpLCZGKUYnRichIiIiIiNGJyw+KiQpRikiIzhGJ0YnKiYiIzxGJylGKSIjN0YnRi0qJiIjaUYnKUYpIiM2RidGJyomIiRBIkYnKUYpIiM1RidGLSomIiRKJkYnKUYpIiIqRidGJyomIiR6J0YnKUYpIiIpRidGLSomIiVDOkYnKUYpIiIoRidGJyomIiVnT0YnKUYpIiInRidGLSomIiU+PEYnKUYpIiImRidGJyomIiVSbUYnKUYpIiIlRidGLSomIiVFOEYnRihGJ0YnKiYiJVlORicpRilGLkYnRi0qJiIkIikqRidGKUYnRiciJEYkRidGJywmRilGJ0YnRidGLSksJiokRmVuRidGJ0YmRidGQkYtRi0qJiNGJ0YmRidJJFE0MEdGKkYnRi0=

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

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

Ky9JInRHNiIiIiIiIiMiIioiIiUiIyMqIiInIiVJNSIiKSImbUEiIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

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

LC4qJiInT0U6IiIiKUkidEc2IiIjN0YlRiUqJiImbUEiRiUpRiciIzVGJUYlKiYiJUk1RiUpRiciIilGJUYlKiYiIyMqRiUpRiciIidGJUYlKiYiIipGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

n:=14: normal(series(subs(Zm=Zmser(n),Q20=coeff(coeff(convert(Qser(n),polynom),x,2),y,0),Cm40sol),t,n+1));

Ky9JInRHNiIiIiMiIiUiI0siIiciJFQlIiIpIiVNZiIjNSImYispIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

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

LCwqJiImYispIiIiKUkidEc2IiIjN0YlRiUqJiIlTWZGJSlGJyIjNUYlRiUqJiIkVCVGJSlGJyIiKUYlRiUqJiIjS0YlKUYnIiInRiVGJSomIiIjRiUpRiciIiVGJUYl

n:=14: normal(series(subs(Zm=Zmser(n),Q40=coeff(coeff(convert(Qser(n),polynom),x,4),y,0),Cm60sol),t,n+1));

Ky9JInRHNiIiIiYiIiciJDwiIiIpIiVePyIjNSImKmZLIiM3IicucVwiIzktSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

coeff(coeff(convert(Cser(n),polynom),x,-6),y,0);

LCwqJiInLnFcIiIiKUkidEc2IiIjOUYlRiUqJiImKmZLRiUpRiciIzdGJUYlKiYiJV4/RiUpRiciIzVGJUYlKiYiJDwiRiUpRiciIilGJUYlKiYiIiZGJSlGJyIiJ0YlRiU=

n:=14: normal(series(subs(Zm=Zmser(n),Q60=coeff(coeff(convert(Qser(n),polynom),x,6),y,0),Cm80sol),t,n+1));

Ky1JInRHNiIiIzkiIikiJE0lIiM1IiVzIyoiIzciJ0EyPEYlLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOw==

coeff(coeff(convert(Cser(n),polynom),x,-8),y,0);

LCoqJiInQTI8IiIiKUkidEc2IiIjOUYlRiUqJiIlcyMqRiUpRiciIzdGJUYlKiYiJE0lRiUpRiciIzVGJUYlKiZGKUYlKUYnIiIpRiVGJQ==

<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;

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYw

C00sol:=P00Zm+1/3*Q00;

LCYqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYwKiYjRidGJkYnSSRRMDBHRipGJ0Yn

n:=12: normal(series(subs(Zm=Zmser(n),Q00=coeff(coeff(convert(Qser(n),polynom),x,0),y,0),C00sol),t,n+1));;

KzFJInRHNiIiIiIiIiEiIiQiIiMiI0EiIiUiJDQjIiInIiVjQSIiKSImJ0hFIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

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

LDAqJiInJ3BBJCIiIilJInRHNiIiIzdGJUYlKiYiJidIRUYlKUYnIiM1RiVGJSomIiVjQUYlKUYnIiIpRiVGJSomIiQ0I0YlKUYnIiInRiVGJSomIiNBRiUpRiciIiVGJUYlKiYiIiRGJSlGJyIiI0YlRiVGJUYl

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

collect(convert(subs(M(0,0)=0,series(convert(series(eqcatM,x,3),polynom),y,4)),polynom),[x,y],distributed);

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

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

LCoqJkkidEc2IiIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlRiYiIiNGLzYjSSJNR0YlNiQiIiFGM0YmISIiKigiIiRGJkYkRiYtLSZGKjYjRiZGMEYyRiZGNC0tJkYqNiNGL0YwRjJGJiooI0YmIiInRiZGJEYmLS0mRio2JUYmRiZGJkYwRjJGJkY0

M12sol:=expand(solve(%,(D[1, 2, 2](M))(0, 0))/2);

LCgqJiMiIiIiIzdGJS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlRiVGJUYlNiNJIk1HRi42JCIiIUYzRiUhIiIqJiMiIiQiIiNGJS0tJkYqNiNGJUYwRjJGJUY0KigjRiVGOEYlSSJ0R0YuRjQtLSZGKjYjRjhGMEYyRiVGJQ==

M12Zm:=factor(subs(Tm=Zm^2,-M30Zm/2-3/2*M10Zm+1/2*M01ftTm/t2Tm));

LCQqLCMiI0siIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMiokKUYpIiIoRidGJyomIiM2RicpRikiIidGJ0YrKiYiIzpGJylGKSIiJkYnRisqJiIjZEYnKUYpIiIlRidGJyomRiZGJylGKUYmRidGJyomIiROIkYnKUYpIiIjRidGJyomIiNWRidGKUYnRidGQ0YnRidGPUYnKSwmKiRGQEYnRidGJkYnRjNGK0Yn

Cm22sol:=M12Zm-1/3*Q02:

n:=14:series(subs(Zm=Zmser(n),Q02=coeff(coeff(convert(Qser(n),polynom),x,0),y,2),Cm22sol),t,n+1);

KzFJInRHNiIiIiIiIiMiIzgiIiUiJGQiIiInIiVPPiIiKSImalgjIiM1Iid0Jz4kIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlIiM5

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

LDAqJiIoNitEJSIiIilJInRHNiIiIzlGJUYlKiYiJ3QnPiRGJSlGJyIjN0YlRiUqJiImalgjRiUpRiciIzVGJUYlKiYiJU8+RiUpRiciIilGJUYlKiYiJGQiRiUpRiciIidGJUYlKiYiIzhGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,1) : value of M21</Text-field>

collect(convert(series(convert(subs(M(0,0)=0,series(eqcatM,x,4)),polynom),y,3),polynom),[x,y],distributed,factor): coeff(coeff(%,x,1),y,0);

LCoqJkkidEc2IiIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlRiZGJiIiIzYjSSJNR0YlNiQiIiFGM0YmISIiKiZGL0YmLS0mRio2I0YmRjBGMkYmRiYqKEYvRiZGJEYmLS0mRio2I0YvRjBGMkYmRjQjRi8iIiRGNA==

M21ftsol:=expand(solve(%,(D[1, 1, 2](M))(0, 0))/2*t);

LCgtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIjYjSSJNR0YqNiQiIiFGMEYsKiZJInRHRipGLC0tJkYmNiMiIiNGLUYvRiwhIiIjRiwiIiRGOA==

M21ftZm:=factor(subs(Tm=Zm^2,M10Zm-M01ftTm-1/3));

LCQqLCMiIiMiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMCokKUYpIiInRidGJyomIiM5RicpRikiIiZGJ0YrKiYiIz5GJylGKSIiJUYnRicqJiIjP0YnKUYpRiZGJ0YnKiYiIzpGJylGKUYlRidGJyomRi9GJ0YpRidGK0YmRitGJyksJiokRj1GJ0YnRiZGJ0YmRissJkYpRidGJ0YnRitGJw==

Cm31ftsol:=M21ftZm-t/3*Q11:

n:=14:series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),Cm31ftsol),t,n+1);

Ky9JInRHNiIiIiMiIiUiI0YiIiciJFEkIiIpIiV2VSIjNSImTV8mIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

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

LC4qJiInMydHKCIiIilJInRHNiIiIzhGJUYlKiYiJk1fJkYlKUYnIiM2RiVGJSomIiV2VUYlKUYnIiIqRiVGJSomIiRRJEYlKUYnIiIoRiVGJSomIiNGRiUpRiciIiZGJUYlKiYiIiNGJSlGJyIiJEYlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,2) : the series P02</Text-field>

limit(PexprM,x=0);

KiYsKComLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiI2I0kiTUdGLDYkIiIhSSJ5R0YsRi5GM0YuRi4tRjA2JEYzRjJGLi1GMDYkRjJGMiEiIkYuRjNGOA==

series(%,y,4);

KytJInlHNiIsJComIiIjIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKDYjSSJNR0YkNiQiIiFGNEYoRihGNCwmLS0mRiw2JEYoRidGMUYzRigqJiNGKEYnRigtLSZGLDYkRihGKEYxRjNGKEYoRigsJiomRjtGKC0tJkYsNiVGKEYnRidGMUYzRihGKComI0YoIiInRigtLSZGLDYlRihGKEYoRjFGM0YoRihGJy1JIk9HRi5GMCIiJA==

P02Zm:=factor(M12Zm+M30Zm);

LCQqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiIjRicpRikiIidGJ0YsKiZGOEYnKUYpIiImRidGLComIiMlKUYnKUYpIiIlRidGJyomIiNVRidGKEYnRicqJiIkYSJGJylGKUY2RidGJyomIiNRRidGKUYnRiciIz5GJ0YnLCZGKUYnRidGJ0YsKSwmKiRGREYnRidGJkYnRjhGLEYn

C02sol:=P02Zm+Q02/3:

n:=14:series(subs(Zm=Zmser(n), Q02=coeff(coeff(convert(Qser(n),polynom),x,0),y,2),C02sol),t,n+1);

KzFJInRHNiIiIiNGJSIjQCIiJSIkTSMiIiciJXBGIiIpIiZ4VSQiIzUiJ0sqUSUiIzctSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

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

LDAqJiIocCdwZCIiIilJInRHNiIiIzlGJUYlKiYiJ0sqUSVGJSlGJyIjN0YlRiUqJiImeFUkRiUpRiciIzVGJUYlKiYiJXBGRiUpRiciIilGJUYlKiYiJE0jRiUpRiciIidGJUYlKiYiI0BGJSlGJyIiJUYlRiUqJiIiI0YlKUYnRj9GJUYl

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

coeff(series(PexprM,x,3),x,1);;

LCgqJiMiIiIiIiNGJS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkRiVGJTYjSSJNR0YuNiQiIiFJInlHRi5GJUYlKiZGNCEiIi0tJkYqNiNGJkYwNiRGNEYzRiVGJSomRjRGNi1GODYkRjNGM0YlRjY=

series(%,y,3);

KylJInlHNiIsJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkIiIiIiIjNiNJIk1HRiQ2JCIiIUYzRi4qJiNGLkYvRi4tLSZGKTYkRi5GLkYwRjJGLkYuRjMtLSZGKTYlRi5GLkYvRjBGMkYuLUkiT0dGKzYjRi5GLw==

coeff(%,y,1);

LS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUiIiJGKyIiIzYjSSJNR0YpNiQiIiFGMA==

P11ftZm:=2*M21ftZm;

LCQqLCMiIiUiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMCokKUYpIiInRidGJyomIiM5RicpRikiIiZGJ0YrKiYiIz5GJylGKUYlRidGJyomIiM/RicpRilGJkYnRicqJiIjOkYnKUYpIiIjRidGJyomRi9GJ0YpRidGK0YmRitGJyksJiokRjxGJ0YnRiZGJ0YmRissJkYpRidGJ0YnRitGJw==

C11ftsol:=P11ftZm+Q11*t/3:

n:=14:series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),C11ftsol),t,n+1);

KzFJInRHNiIiIiIiIiMiIikiIiUiI3oiIiciJHMpRiciJjkuIiIjNSInIyp5NyIjNy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjOQ==

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

LDAqJiIoZDdrIiIiIilJInRHNiIiIzhGJUYlKiYiJyMqeTdGJSlGJyIjNkYlRiUqJiImOS4iRiUpRiciIipGJUYlKiYiJHMpRiUpRiciIihGJUYlKiYiI3pGJSlGJyIiJkYlRiUqJiIiKUYlKUYnIiIkRiVGJUYnRiU=

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

I have to restart Maple, otherwise hypergeomsols bugs...

restart:

with(gfun): with(LREtools):

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:

Zmser(10);

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

<Text-field style="Heading 3" layout="Heading 3">First, the coefficients of T and Z</Text-field>

de:=(algeqtodiffeq(subs(t=sqrt(s),algTm),Tm(s)));

PCQsLCIjWyEiIiomLCYqJiIlIUciIiIiSSJzRzYiRipGJSIjU0YqRiotSSNUbUdGLDYjRitGKkYqKiYsKComRilGKilGKyIiI0YqRioqJiIjISlGKkYrRipGJUY1RipGKi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGLkYrRipGKiomLCgqJiIlOyMqRiopRisiIiRGKkYqKiYiJGspRipGNEYqRiUqJiIjPUYqRitGKkYqRiotRjk2JEY4RitGKkYqKiYsKComIiUvQkYqKUYrIiIlRipGKiomIiQpR0YqRkBGKkYlKiYiIipGKkY0RipGKkYqLUY5NiRGRkYrRipGKi8tRi82IyIiIS1JJ1Jvb3RPZkc2JEY6SShfc3lzbGliR0YsNiMsKCokKUkjX1pHRlpGNUYqRioqJkY1RipGam5GKkYqRkFGJQ==

de0:={op(1,de), Tm(0)=1}:

rec:=diffeqtorec(de0,Tm(s),co(k));

PCUsKComLCoqJiIlL0IiIiIpSSJrRzYiIiIkRihGKComRidGKClGKiIiI0YoRigqJiIlR0xGKEYqRighIiIiJSFHIkYyRigtSSNjb0dGKzYjRipGKEYoKiYsKiomIiQpR0YoRilGKEYyKiYiJGspRihGLkYoRjIqJiIkYydGKEYqRihGMiIjU0YyRigtRjU2IywmRipGKEYoRihGKEYoKiYsKiomIiIqRihGKUYoRigqJiIjWEYoRi5GKEYoKiYiI3VGKEYqRihGKEY/RihGKC1GNTYjLCZGKkYoRi9GKEYoRigvLUY1NiMiIiFGKC8tRjU2I0YoIiIl

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

LCgqLCIkYyMiIiIsJiomIiIkRiVJImtHNiJGJUYlIiImRiVGJSwmRilGJUYlISIiRiUsJkYnRiVGJUYlRiUtSSNjb0dGKjYjRilGJUYlKigiIilGJSwqKiYiI09GJSlGKUYoRiVGJSomIiQzIkYlKUYpIiIjRiVGJSomIiMjKUYlRilGJUYlRitGJUYlLUYwNiMsJkYpRiVGJUYlRiVGLSoqRiZGJSwmRidGJSIiJUYlRiUsJkYpRiVGO0YlRiUtRjA2I0ZERiVGJQ==

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

PCQvLUkjY29HNiI2IyIiISIiIi8tRiU2I0YpIiIl

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

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

IiIh

de:=(algeqtodiffeq(subs(t=sqrt(s),algZm),Zm(s)));

PCQsLComIiRTIyIiIi1JI1ptRzYiNiNJInNHRilGJiEiIiomLCYqJiIlK2tGJkYrRiZGJiIkKyNGLEYmLUklZGlmZkclKnByb3RlY3RlZEc2JEYnRitGJkYmKiYsKComIiZbMyRGJilGKyIiI0YmRiYqJiIlRz5GJkYrRiZGLCIjP0YmRiYtRjM2JEYyRitGJkYmKiYsKComIiZLJT1GJilGKyIiJEYmRiYqJiIlRzxGJkY6RiZGLComIiNPRiZGK0YmRiZGJi1GMzYkRj9GK0YmRiYqJiwoKiYiJS9CRiYpRisiIiVGJkYmKiYiJClHRiZGRUYmRiwqJiIiKkYmRjpGJkYmRiYtRjM2JEZLRitGJkYmLy1GKDYjIiIhLUknUm9vdE9mRzYkRjRJKF9zeXNsaWJHRik2IywoKiQpSSNfWkdGaW5GUkYmRiYqJkY7RiYpRl9vRjtGJkYmRkZGLA==

de0:={op(1,de), Zm(0)=1}:

rec:=diffeqtorec(de0,Zm(s),co(k));

PCQsKComLCwqJiIlL0IiIiIpSSJrRzYiIiIlRihGKComIiUzWUYoKUYqIiIkRihGKComIiQnKilGKClGKiIiI0YoRigqJiIlMzlGKEYqRighIiIiJFMjRjdGKC1JI2NvR0YrNiNGKkYoRigqJiwsKiYiJClHRihGKUYoRjcqJiIlXzZGKEYvRihGNyomIiVTO0YoRjNGKEY3KiYiJHcqRihGKkYoRjciJCsjRjdGKC1GOjYjLCZGKkYoRihGKEYoRigqJiwsKiYiIipGKEYpRihGKComIiNhRihGL0YoRigqJiIkPiJGKEYzRihGKComIiQ5IkYoRipGKEYoIiNTRihGKC1GOjYjLCZGKkYoRjRGKEYoRigvLUY6NiMiIiFGKA==

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

LCgqLiIjOyIiIiwmKiYiIidGJUkia0c2IkYlRiUiIiZGJUYlLCYqJiIiI0YlRilGJUYlIiIkRiVGJSwmRi1GJUYlISIiRiUsJkYnRiVGJUYlRiUtSSNjb0dGKjYjRilGJUYlKioiIilGJSwoKiYiI09GJSlGKUYuRiVGJSomIiNzRiVGKUYlRiUiI0RGJUYlKSwmRilGJUYlRiVGLkYlLUY0NiNGQEYlRjEqLCwmKiZGL0YlRilGJUYlRitGJUYlLCZGRUYlIiIlRiVGJSwmRilGJUYuRiVGJUZARiUtRjQ2I0ZIRiVGJQ==

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

PCMvLUkjY29HNiI2IyIiISIiIg==

hypergeomsols(rec1,co(k),{co(0) = 1,co(1)=2,co(2)=16});

LCYqNCMiIiIiIiNGJSlJI1BpRyUqcHJvdGVjdGVkR0YkISIiLUkmR0FNTUFHNiRGKUkoX3N5c2xpYkc2IjYjIyIiJiIiJ0YqLUYsNiMjRiYiIiRGJSkiIztJImtHRi9GJS1GLDYjLCZGOkYlRiRGKkYlLUYsNiMsJkY6RiVGMUYlRiUtRiw2IywmRjpGJUYlRiVGKi1GLDYjLCZGOkYlRjZGJUYqRiUqNiNGJUY3RiUpRjdGJEYlRidGKkY0RipGK0YlRjhGJUY7RiUtRiw2IywmRjpGJSNGJUYzRiVGJUZBRiotRiw2IywmRjpGJUZIRiVGKkYq

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

seq(ifactor((1/2)*GAMMA(2/3)*16^k*GAMMA(k+5/6)*GAMMA(k-1/2)/(sqrt(Pi)*GAMMA(5/6)*GAMMA(k+1)*GAMMA(k+2/3))/ (16^k*3/4*fact(1/2,k-1)*fact(5/6,k)/fact(2,k-1)/fact(5/3,k-1))),k=1..10);

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

seq(ifactor((1/3)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k-1/2)*GAMMA(k+1/6)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+1/3)*GAMMA(k+1))/ (16^k/2*fact(1/2,k-1)*fact(7/6,k-1)/fact(2,k-1)/fact(4/3,k-1))),k=1..10);

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

pap:=proc(n) 16^n*( 2* fact(-1/2,n)*fact(1/6,n)/fact(1,n)/fact(1/3,n)-fact(-1/2,n)*fact(5/6,n)/fact(1,n)/fact(2/3,n)): end:

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

Ni0iIiIiIiMiIzsiJG0iIiVNPiImd1MjIicrSEoiKEc6PiUiKVtbVWQiKkc+XispIiw3dmg5OCI=

Zmser(20);

KzlJInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpIiZ3UyMiIzUiJytISiIjNyIoRzo+JSIjOSIpW1tVZEYoIipHPl4rKSIjPS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjPw==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at abscissa -1: not HG</Text-field>

M01ftTm:=-(Tm-1)*(Tm^2-6*Tm-11)/(Tm+3)^3;

LCQqKCwmSSNUbUc2IiIiIkYnISIiRicsKCokKUYlIiIjRidGJyomIiInRidGJUYnRigiIzZGKEYnKSwmRiVGJyIiJEYnRjJGKEYo

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algTm,Tm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

LCwqLCImT2InIiIiSSJrRzYiRiUsJiomIiIkRiVGJkYlRiUiIiNGJUYlLCZGKUYlRiVGJUYlLUkjY29HRic2I0YmRiVGJSooIiVbP0YlLCgqJiIjYUYlKUYmRitGJUYlKiYiJEMiRiVGJkYlRiUiI2xGJUYlLUYuNiMsJkYmRiVGJUYlRiVGJSoqIiRvKEYlLCYqJkYrRiVGJkYlRiUiIiZGJUYlLCgqJkYqRiVGNUYlRiUqJiIjOkYlRiZGJUYlIiM4RiVGJS1GLjYjLCZGJkYlRitGJUYlISIiKigiIilGJSwoRjNGJSomIiQ7JUYlRiZGJUYlIiQmekYlRiUtRi42IywmRiZGJUYqRiVGJUZJKiosJkYmRiVGQEYlRiUsJkYpRiUiIzlGJUYlLCZGKUYlRkVGJUYlLUYuNiMsJkYmRiUiIiVGJUYlRiU=

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRiwvLUYlNiMiIiMiIigvLUYlNiMiIiQiI2w=

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

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

IiIh

M03ftTm:=(Tm^4-44*Tm^3+110*Tm^2+420*Tm+281)*(Tm-1)^2/(Tm+3)^6;

KigsLCokKUkjVG1HNiIiIiUiIiJGKSomIiNXRikpRiYiIiRGKSEiIiomIiQ1IkYpKUYmIiIjRilGKSomIiQ/JUYpRiZGKUYpIiQiR0YpRikpLCZGJkYpRilGLkYyRikpLCZGJkYpRi1GKSIiJ0Yu

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algTm,Tm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

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

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

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMiIiQvLUYlNiNGMSIjVy8tRiU2IyIiJSIkImUvLUYlNiMiIiYiJUd3Ly1GJTYjIiInIic+ODU=

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

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

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0), (-4,0), (-6,0), (-8,0) : hypergeometric</Text-field>

M10Zm := -(32/3)*Zm^3*(Zm^3-3*Zm^2-Zm-1)/((Zm^2+3)^3*(Zm+1));

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYw

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

PCYsKiomLCwqJiIma28kIiIiKUkia0c2IiIiJUYoRigqJiInY3U5RigpRioiIiRGKEYoKiYiJ1c0QEYoKUYqIiIjRihGKComIid3cDdGKEYqRihGKCImISlvI0YoRigtSSNjb0dGKzYjRipGKEYoKiYsLComIiU3cEYoRilGKCEiIiomIiYpb11GKEYvRihGPyomIidLKzlGKEYzRihGPyomIicpR3MiRihGKkYoRj8iJlMlekY/RigtRjk2IywmRipGKEYoRihGKEYoKiYsLComIiRLJUYoRilGKEYoKiYiJTNZRihGL0YoRigqJiImcyU9RihGM0YoRigqJiImM0kkRihGKkYoRigiJitBI0YoRigtRjk2IywmRipGKEY0RihGKEYoKiYsLComIiIqRihGKUYoRj8qJiIkRSJGKEYvRihGPyomIiRmJ0YoRjNGKEY/KiYiJUU6RihGKkYoRj8iJT84Rj9GKC1GOTYjLCZGKkYoRjBGKEYoRigvLUY5NiMiIiEjRihGMC8tRjk2I0YoI0YsRjAvLUY5NiNGNCMiI0pGMA==

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

LCoqLiIkYyMiIiIsJiomIiInRiVJImtHNiJGJUYlIiImRiVGJSwmKiYiIiNGJUYpRiVGJSIiJEYlRiUsJkYtRiVGJUYlRiUsJkYnRiUiIihGJUYlLUkjY29HRio2I0YpRiVGJSoqIiM7RiVGLEYlLCoqJiIkOyNGJSlGKUYvRiVGJSomIiVnN0YlKUYpRi5GJUYlKiYiJSdbI0YlRilGJUYlIiViO0YlRiUtRjQ2IywmRilGJUYlRiVGJSEiIioqIiIpRiUsJkYpRiVGL0YlRiUsKiomIiNhRiVGO0YlRiUqJiIkOSVGJUY+RiVGJSomIiVuNUYlRilGJUYlIiREKkYlRiUtRjQ2IywmRilGJUYuRiVGJUYlKiwsJiomRi9GJUYpRiVGJSIjNUYlRiUsJkYpRiUiIiVGJUYlRkhGJSwmRlZGJSIjNkYlRiUtRjQ2I0ZIRiVGRQ==

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

PCUvLUkjY29HNiI2IyIiISMiIiIiIiQvLUYlNiNGKiMiIiVGKy8tRiU2IyIiIyMiI0pGKw==

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

LCYqNiMiIikiI0YiIiIpIiIkI0YnIiIjRicpSSNQaUclKnByb3RlY3RlZEdGKiEiIi1JJkdBTU1BRzYkRi5JKF9zeXNsaWJHNiI2IyNGK0YpRi8tRjE2IyMiIiYiIidGJykiIztJImtHRjRGJy1GMTYjLCZGPkYnIyIiKEY7RidGJy1GMTYjLCZGPkYnRipGJ0YnLUYxNiMsJkY+RicjIiIlRilGJ0YvLUYxNiMsJkY+RidGK0YnRi9GJyo0I0YrRiZGJ0YsRi9GN0YvRjBGJ0Y8RictRjE2IywmRj5GJ0Y5RidGJ0ZERidGTEYvLUYxNiMsJkY+RicjRjpGKUYnRi9GLw==

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

eqpap:=proc(n) 16^n/9*(-3*fact(1/2,n)^2/fact(2,n)^2+4*fact(1/2,n)*fact(7/6,n)/fact(2,n)/fact(4/3,n) -fact(1/2,n)*fact(5/6,n)/fact(2,n)/fact(5/3,n)): end:

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

Ni0iIiEiIiIiIioiIyMqIiVJNSImbUEiIidPRToiKFhEJz4iKSUpKnllIyIqITNrIlskIisoZks1dyU=

n:=20:coeff(coeff(convert(Cser(n),polynom),x,-2),y,0);

LDYqJiIrKGZLNXclIiIiKUkidEc2IiIjP0YlRiUqJiIqITNrIlskRiUpRiciIz1GJUYlKiYiKSUpKnllI0YlKUYnIiM7RiVGJSomIihYRCc+RiUpRiciIzlGJUYlKiYiJ09FOkYlKUYnIiM3RiVGJSomIiZtQSJGJSlGJyIjNUYlRiUqJiIlSTVGJSlGJyIiKUYlRiUqJiIjIypGJSlGJyIiJ0YlRiUqJiIiKkYlKUYnIiIlRiVGJSokKUYnIiIjRiVGJQ==

M30Zm := (32/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+22*Zm^6-54*Zm^5+108*Zm^4-54*Zm^3+130*Zm^2-10*Zm-5)/((Zm+1)*(Zm^2+3)^6);

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

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

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

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIiIlLy1GJTYjRi4iI1ovLUYlNiNGMyMiJTo8Ri4vLUYlNiMiIiYiJSU+KC8tRiU2IyIiJyImQkoq

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

LCYqOCMiIikiJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJykiIztJImtHRjFGJy1GLTYjLCZGOEYnIyIiKEY1RidGJy1GLTYjLCZGOEYnRipGJ0YnLCgqJiIjQEYnKUY4RitGJ0YnKiYiI0lGJ0Y4RidGJyIjOSEiIkYnKUkjUGlHRi9GKkZILUYtNiMjRitGKUZILUYtNiMsJkY4RicjRj1GKUYnRkgtRi02IywmRjhGJ0YpRidGSEYnKjYjRkdGJkYnRktGJ0Y2RictRi02IywmRjhGJ0YzRidGJ0Y+RicsKComRilGJ0ZERidGJyomRilGJ0Y4RidGJyIjNUZIRidGLEZIRklGSC1GLTYjLCZGOEYnI0YlRilGJ0ZIRlJGSEZI

eqpap:=proc(n) 16^n/3^5*(-3^5*n*fact(1/2,n)^2/fact(2,n)/fact(2,n+1)+4*(21*n^2+30*n-14)*fact(1/2,n)*fact(7/6,n)/fact(2,n+1)/fact(4/3,n+1) -7*(3*n^2+3*n-10)*fact(1/2,n)*fact(5/6,n)/fact(2,n+1)/fact(5/3,n+1)): end:

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

Ni0iIiFGIyIiIyIjSyIkVCUiJU1mIiZiKykiKEgtNCIiKTEiM10iIipiKG8oMyMiK0tmLEtI

n:=20:coeff(coeff(convert(Cser(n),polynom),x,-4),y,0);

LDQqJiIrS2YsS0giIiIpSSJ0RzYiIiM/RiVGJSomIipiKG8oMyNGJSlGJyIjPUYlRiUqJiIpMSIzXSJGJSlGJyIjO0YlRiUqJiIoSC00IkYlKUYnIiM5RiVGJSomIiZiKylGJSlGJyIjN0YlRiUqJiIlTWZGJSlGJyIjNUYlRiUqJiIkVCVGJSlGJyIiKUYlRiUqJiIjS0YlKUYnIiInRiVGJSomIiIjRiUpRiciIiVGJUYl

M50Zm := -(32/3)*Zm^3*(Zm-1)^2*(Zm^13-17*Zm^12+62*Zm^11-122*Zm^10+531*Zm^9-679*Zm^8+1524*Zm^7-3660*Zm^6+1719*Zm^5-6639*Zm^4+1326*Zm^3-3546*Zm^2+981*Zm+327)/((Zm+1)*(Zm^2+3)^9);

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

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

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMjRjAiIiQvLUYlNiNGMiMiI1NGMi8tRiU2IyIiJSMiJEonRjIvLUYlNiMiIiYiJSxKLy1GJTYjIiInIiYqcFcvLUYlNiMiIigiJ1ksay8tRiU2IyIiKSMiKW9ZXUZGMi8tRiU2IyIiKiIqX0dvSiI=

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

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

M70Zm := (32/3)*Zm^3*(Zm-1)^3*(Zm^18-24*Zm^17+119*Zm^16-132*Zm^15+1088*Zm^14-3428*Zm^13+3112*Zm^12-26004*Zm^11+18378*Zm^10-40356*Zm^9+130542*Zm^8+6516*Zm^7+303144*Zm^6+21396*Zm^5+270672*Zm^4-44124*Zm^3+70605*Zm^2-44916*Zm-11229)/((Zm+1)*(Zm^2+3)^12);

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

PC8vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCMiIiZGNC8tRiU2IyIiJSMiJFMiRjQvLUYlNiNGNiIkQyovLUYlNiMiIiciJltnIi8tRiU2IyIiKCInOD1FLy1GJTYjIiIpIyIpJFEkUjdGNC8tRiU2IyIiKiIpV1onUicvLUYlNiMiIzUiKiYqXGJ6Ki8tRiU2IyIjNiIseikpXC9cIi8tRiU2IyIjNyMiLSlSZnIleW5GNA==

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

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

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

P00Zm := -(64/3)*Zm^3*(Zm^3-3*Zm^2-Zm-1)/((Zm^2+3)^3*(Zm+1));

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnKSwmKiRGLkYnRidGJkYnRiZGMCwmRilGJ0YnRidGMEYw

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de:=map(factor,algeqtodiffeq(alg,G(s))):

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

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

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

PCUvLUkjY29HNiI2IyIiISMiIiMiIiQvLUYlNiMiIiIjIiIpRisvLUYlNiNGKiMiI2lGKw==

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

LCYqNiMiIzsiI0YiIiIpIiIkI0YnIiIjRicpSSNQaUclKnByb3RlY3RlZEdGKiEiIi1JJkdBTU1BRzYkRi5JKF9zeXNsaWJHNiI2IyNGK0YpRi8tRjE2IyMiIiYiIidGJylGJUkia0dGNEYnLUYxNiMsJkY9RidGKkYnRictRjE2IywmRj1GJyMiIihGO0YnRictRjE2IywmRj1GJ0YrRidGLy1GMTYjLCZGPUYnIyIiJUYpRidGL0YnKjQjRk1GJkYnRixGL0Y3Ri9GMEYnRjxGJy1GMTYjLCZGPUYnRjlGJ0YnRj5GJy1GMTYjLCZGPUYnI0Y6RilGJ0YvRkZGL0Yv

eqpap:=proc(n) 16^n/9*(3*fact(1/2,n)^2/fact(2,n)^2+8*fact(1/2,n)*fact(7/6,n)/fact(2,n)/fact(4/3,n) -2*fact(1/2,n)*fact(5/6,n)/fact(2,n)/fact(5/3,n)): end:

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

Ni0iIiIiIiQiI0EiJDQjIiVjQSImJ0hFIicncEEkIihKIjRUIilvRyFRJiIqL3MnKj4oIis1MzwvKSo=

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

LDgqJiIrNTM8LykqIiIiKUkidEc2IiIjP0YlRiUqJiIqL3MnKj4oRiUpRiciIz1GJUYlKiYiKW9HIVEmRiUpRiciIztGJUYlKiYiKEoiNFRGJSlGJyIjOUYlRiUqJiInJ3BBJEYlKUYnIiM3RiVGJSomIiYnSEVGJSlGJyIjNUYlRiUqJiIlY0FGJSlGJyIiKUYlRiUqJiIkNCNGJSlGJyIiJ0YlRiUqJiIjQUYlKUYnIiIlRiVGJSomIiIkRiUpRiciIiNGJUYlRiVGJQ==

The dominant term is the one with Gamma(k+7/6)

map(simplify,expand(asympt((16/27)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+1/2)*GAMMA(k+7/6)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+2)*GAMMA(k+4/3)),k,2)));

LCYqMCMiIikiIioiIiIpIiM7SSJrRzYiRicpIiIjI0YnIiIkRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjI0YtRi9GJ0kjUGlHRjMhIiJGKkY4KSokRipGOEY2RidGJyomRihGJy1JIk9HRjM2IyomKUYqRi1GOEY5RidGJ0Yn

map(simplify,expand(asympt((4/27)*GAMMA(2/3)*16^k*GAMMA(k+5/6)*GAMMA(k+1/2)/(sqrt(Pi)*GAMMA(5/6)*GAMMA(k+5/3)*GAMMA(k+2)),k,3)));

LCYqMCMiIiUiIyIpIiIiKSIjO0kia0c2IkYnKSIiIyNGLSIiJEYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2I0YuISIiKUYvI0YnRi1GJylGKkYtRjYpKiRGKkY2I0YnRi9GJ0YnKiZGKEYnLUkiT0dGMzYjKiYpRipGL0Y2RjpGJ0YnRic=

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

M12Zm := (32/3)*(Zm-1)*(Zm^7-11*Zm^6-15*Zm^5+57*Zm^4+3*Zm^3+135*Zm^2+43*Zm+43)*Zm^3/(Zm^2+3)^6;

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de0:=algeqtodiffeq(alg,G(s)):nops(de0);

IiIj

op(2,de0);

Ly1JIkdHNiI2IyIiIS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqJClJI19aR0YqIiIjIiIiRjNGMSEiIg==

n:=4: series(subs(Zm=Zmser(n),M12Zm),t,n+1);

KydJInRHNiIjIiIlIiIkIiIjLUkiT0clKnByb3RlY3RlZEc2IyIiIkYm

de:={op(1,de0), G(0)=0}:

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

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

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

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJSIiJC8tRiU2IyIiIyIjOi8tRiU2I0YvIiRzIi8tRiU2I0YuIyIlK2lGLy8tRiU2IyIiJiImQmUjLy1GJTYjIiInIidURkwvLUYlNiMiIigiKGFKUiUvLUYlNiMiIikiKSJmXCJmLy1GJTYjIiIqIipVLkY0KS8tRiU2IyIjNSMiLDUuSWpPJEYvLy1GJTYjIiM2Ii1zY0JgdDovLUYlNiMiIzciLjpPRFt6QSMvLUYlNiMiIzgjIi93PDJOJUdhKkYvLy1GJTYjIiM5IjA/I1wnKWZ4dVg=

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

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

LCYqOiMiJHMjIiUuXiIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRictRi02IyNGK0YpISIiLCYqJiIjN0YnLUkjY29HRjE2I0YnRidGJyomIiNKRictRj42IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGSUYnIyIiKEY1RidGJy1GLTYjLCZGSUYnRipGJ0YnLUYtNiMsJkZJRicjRk5GKUYnRjktRi02IywmRklGJ0YpRidGOSwoKiQpRklGK0YnRicqJiMiI10iIzxGJ0ZJRidGJyMiJCwiIiNeRidGJ0YnKjgjIiNvIiY6YiNGJ0Y2RicsJiomIiQuJEYnRj1GJ0YnKiYiJTM3RidGQkYnRjlGJ0ZFRjlGLEY5RkdGJy1GLTYjLCZGSUYnRjNGJ0YnRk9GJy1GLTYjLCZGSUYnIyIiKUYpRidGOUZWRjksKEZaRicqJiMiI2BGaW5GJ0ZJRidGJyMiJCsiRlxvRidGJ0Y5

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

M21ftZm := (2/3)*(Zm-1)*(Zm^6-14*Zm^5+19*Zm^4+20*Zm^3+15*Zm^2-6*Zm-3)/((Zm^2+3)^3*(Zm+1));

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de0:=algeqtodiffeq(alg,G(s)):nops(de0);

IiIj

op(2,de0);

Ly1JIkdHNiI2IyIiIUYn

de:=de0:

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

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

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

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

n:=6: series(subs(Zm=Zmser(n),M21ftZm),t,n+1);

KylJInRHNiIjIiIiIiIkIiIjIyIjNUYnIiIlLUkiT0clKnByb3RlY3RlZEc2I0YmIiIn

subs(_C[0]=1/3,%%%);

PC0vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIyIjNUYuLy1GJTYjRi4jIiQxIkYuLy1GJTYjIiIlIyIlNTdGLi8tRiU2IyIiJiIlalsvLUYlNiMiIiciJlU1Jy8tRiU2IyIiKCInYioqeS8tRiU2IyIiKSMiKSV5MjkkRi4vLUYlNiMiIiojIiovaERDJUYuLy1GJTYjRjQjIitJImUzI2VGLg==

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

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmKiZGK0YnLUkjY29HRjE2IyIiIUYnRictRj02I0YnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnIyIiKEY1RidGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUY5KjYjRitGJkYnRjZGJywmKiZGS0YnRjxGJ0YnKiZGK0YnRkBGJ0Y5RidGQkY5RixGOUZERictRi02IywmRkZGJ0YzRidGJ0ZMRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGJw==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,2) : no</Text-field>

P02Zm := (64/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7-2*Zm^6-6*Zm^5+84*Zm^4+42*Zm^3+154*Zm^2+38*Zm+19)/((Zm+1)*(Zm^2+3)^6);

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de0:=algeqtodiffeq(alg,G(s)):nops(de0);

IiIj

op(2,de0);

Ly1JIkdHNiI2IyIiIUYn

de:=de0:

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

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

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

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJiIiJC8tRiU2IyIiIyIjPi8tRiU2I0YvIiQ+Iy8tRiU2IyIiJSMiJTp6Ri8vLUYlNiNGLiImPEkkLy1GJTYjIiInIidrZVUvLUYlNiMiIigiKEVsaSYvLUYlNiMiIikiKTxPenYvLUYlNiMiIioiK0wsUVA1Ly1GJTYjIiM1IyIsWUBoa0olRi8vLUYlNiMiIzYiLWxRTjc9Py8tRiU2IyIjNyIuXCtfenomRy8tRiU2IyIjOCMiME9OJVtHTEM3Ri8vLUYlNiMiIzkiMDA+TlZQLChl

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

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

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

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

P11ftZm := (4/3)*(Zm-1)*(Zm^6-14*Zm^5+19*Zm^4+20*Zm^3+15*Zm^2-6*Zm-3)/((Zm^2+3)^3*(Zm+1));

alg:=factor(subs(t=sqrt(s),factor(resultant(numer(G-%),algZm,Zm)))):

de0:=algeqtodiffeq(alg,G(s)):nops(de0);

IiIj

op(2,de0);

Ly1JIkdHNiI2IyIiIUYn

de:=de0:

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

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

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

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

n:=6: series(subs(Zm=Zmser(n),P11ftZm),t,n+1);

KylJInRHNiIjIiIjIiIkRiYjIiM/RiciIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

subs(_C[0]=2/3,%%%);

PC0vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiIyIiJC8tRiU2I0YuIyIjP0YvLy1GJTYjRi8jIiQ3I0YvLy1GJTYjIiIlIyIlP0NGLy8tRiU2IyIiJiIlRSgqLy1GJTYjIiInIiclM0EiLy1GJTYjIiIoIig1Kno6Ly1GJTYjIiIpIyIpb2IiRydGLy8tRiU2IyIiKiMiKjNBXlspRi8vLUYlNiMiIzUjIixnaXJUOyJGLw==

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

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnIyIiKEY1RidGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGS0YnRj9GJ0Y5RidGQkY5RixGOUZERictRi02IywmRkZGJ0YzRidGJ0ZMRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

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

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

LCoqJiwmIiIiRiUqJiIiJUYlSSJ0RzYiRiUhIiJGJS1JIkNHRik2JEYlRiVGJUYqRiVGJSooRidGJUYoRiUtSSNDbUdGKTYjRiVGJUYqKiZGKEYlSSNDMEdGKUYlRio=

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

degree(algRsZm,Rsx);

IiIk

res:=factor(subs(x=1,Rsx=Rs1,algRsZm));

LCQqKCIjSyIiIiwqKigiI1tGJUkkUnMxRzYiRiUpSSNabUdGKiIiJEYlRiUqJClGLCIiJUYlRiUqJiIiJ0YlKUYsIiIjRiUhIiIiIihGNUYlKSwoKigiI0NGJUYpRiVGK0YlRiVGLkY1RiVGJUY0RiVGJQ==

n:=10: series(subs(Rs1=subs(x=1,Rsser(n)),Zm=Zmser(n),op(2,res)),t,n+1); series(subs(Rs1=subs(x=1,Rsser(n)),Zm=Zmser(n),op(3,res)),t,n+1);

Ky1JInRHNiIhIzciIiEiI1siIiUiJGspIiInIiYjUjgiIiktSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM/

res1:=factor(resultant(op(1,op(3,res)),algZm,Zm)): nops(res1);

IiM4

(res1);

LDwqJiIxM2kjKjNDaG1AIiIiKUkkUnMxRzYiIiIpRiUhIiIqKCIwd0QuVmMlZiYpRiUpRiciIidGJSlJInRHRigiIiNGJUYlKigiLztzcEk4elhGJSlGJyIiJUYlKUYwRjVGJUYlKigiLSFHOFQ0dChGJSlGJ0YxRiUpRjBGLkYlRiUqJiIrJ0huXEglRiUpRjBGKUYlRiUqJiIvb1pNLCRbbiNGJUYtRiVGKiooIi53Jm8iZT4nR0YlRjRGJUYvRiVGKiooIiw/SmR4QyhGJUY5RiVGNkYlRioqJiIqNzQob2BGJUY6RiVGKiomIi13VEp2KzZGJUY0RiVGKiooIitbd3lUakYlRjlGJUYvRiVGJSomIik7c3g7RiVGNkYlRiUqJiIqV1wqNDpGJUY5RiVGKg==

ifactor([coeffs(%)]);

Ny8sJComKS1JIUc2IjYjIiIjIiNDIiIiKS1GJzYjIiIkIiM8RiwhIiIqJilGJiIjSEYsKUYuIiM4RiwqKEY0RiwpRi4iIilGLC1GJzYjRjdGLCooKUYmIiNNRiwpRi5GKkYsLUYnNiMiIiZGLCokKUYmIiNLRiwsJComRiVGLEY2RixGMiwkKigpRiYiI0RGLEY5RixGO0YsRjIsJCooRjRGLClGLkYwRixGQUYsRjIsJCokRjRGLEYyLCQqJkYlRixGOUYsRjIqKEZLRixGT0YsLUYnNiMiIihGLCokRiVGLCwkKiZGJUYsRkBGLEYy

algRs1:=collect(res1/2^24,Rs1,factor);

LCwqJiIqaiw5SCIiIiIpSSRSczFHNiIiIilGJSEiIiomLCYqJiIpTyQ9NSZGJSlJInRHRigiIiNGJUYlIihCVmYiRipGJSlGJyIiJ0YlRiUqJiwoKiYiKHckSEZGJSlGMCIiJUYlRiUqJiInJ2VxIkYlRi9GJUYqIiVobEYqRiUpRidGOkYlRiUqKiIiKkYlLCYqJiIjS0YlRi9GJUYlRiVGKkYlLCgqJiIkZyJGJUY5RiVGJSomIiM1RiVGL0YlRipGJUYlRiUpRidGMUYlRiUqKEY5RiUpLCYqJkY6RiVGMEYlRiVGJUYqRjFGJSksJkZNRiVGJUYlRjFGJUYl

Rs1 is in fact an even series in t

algRs1s:=collect(expand(subs(t=sqrt(s),algRs1)),Rs1,factor);

LCwqJEkkUnMxRzYiIiIpISpqLDlIIiomLCZJInNHRiUiKU8kPTUmIShCVmYiIiIiRi1GJCIiJ0YtKiYsKCokRioiIiMiKHckSEZGKiEnJ2VxIiElaGxGLUYtRiQiIiVGLSooLCZGKiIjSyEiIkYtRi0sKEYxIiRnIkYqISM1Ri1GLUYtRiRGMiIiKiomRipGMiwmRioiIztGOkYtRjJGLQ==

with(gfun):

Locating possible singularities

factor(discrim(algRs1s,Rs1));

LCQqKEkic0c2IiIiJywmRiQiIzshIiIiIiJGJiwoKiRGJCIiIyIlKVsmRiQhJFYkIiIpRioiIiUhZ25fJno1JypISylvZV9QcydIJ1teYU0iZilHajkuRSp5W01gIg==

{fsolve(%)};

PCQkIiIhRiQkIisrKytdaSEjNg==

algeqtoseries(subs(s=(1-epsilon)/16,algRs1s),epsilon,Rs1,2,true);

NyQsKC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJEkjX1pHRiYiIiMiJFYjISIiIiIiRjEqJi1GJTYjLCYqJEYtIiIkIiRXIkYkRjFGMUkoZXBzaWxvbkdGKSNGMUY3RjEtSSJPR0YnNiMqJEY5I0YuRjdGMSspRjktRiU2IywmRiwiJS9CRjFGMUYxLCRGQSMiIiYiIztGLi1GPDYjRjFGNw==

algeqtoseries(subs(t=(1-epsilon)/4,algRs1),epsilon,Rs1,2,true);

NyQsKC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJEkjX1pHRiYiIiMiJFYjISIiIiIiRjEqJi1GJTYjLCYqJEYtIiIkIiNzRiRGMUYxSShlcHNpbG9uR0YpI0YxRjdGMS1JIk9HRic2IyokRjkjRi5GN0YxKylGOS1GJTYjLCZGLCIkdyZGMUYxRjEsJEZBI0YxIiIpRi4tRjw2I0YxRjc=

sqrt(243);

LCQqJCIiJCMiIiIiIiMiIio=

allvalues(RootOf(72*_Z^3+RootOf(243*_Z^2-1)));

NigsJiokIiIkIyIiIiIiIyNGJyIjT14jI0YnIiM3RicsJEYkIyEiIiIjPSwmRiRGKV4jI0YwRi1GJywkRiQjRidGMSwmRiQjRjBGKkYrRicsJkYkRjhGM0Yn

with(plots):

implicitplot(algRs1s,s=-1/16..1/16.1,Rs1=-0.05..0.05,numpoints=10000);

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

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

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Ig==

evalf(8/9*sqrt(3)/GAMMA(2/3));

JCIrR28ocDgiISIq

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=4..n)],style=point);

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Ig==

evalf(2^5/9*GAMMA(2/3)/Pi);

JCIrMiFbRGAiISIq

<Text-field style="Heading 1" layout="Heading 1">Starting at (-2,0)</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^2-t/y*(x+1/x)*Ch(1/x)-t/x*(y+1/y)*Cv(1/y)-t/x/y*C00;

LCwqJiwmIiIiRiUqKEkidEc2IkYlLCZJInhHRihGJSokRiohIiJGJUYlLCZJInlHRihGJSokRi5GLEYlRiVGLEYlLUkiQ0dGKDYkRipGLkYlRiwqJClGKiIiI0YsRiUqKkYnRiVGLkYsRilGJS1JI0NoR0YoNiNGK0YlRiwqKkYnRiVGKkYsRi1GJS1JI0N2R0YoNiNGL0YlRiwqKkYnRiVGKkYsRi5GLEkkQzAwR0YoRiVGLA==

C00ser:=proc(n) coeff(coeff(convert(Cser(n),polynom),y,0),x,0): end:

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^2 else normal(series(subs(C(x,y)=Cser(n-1),Ch(1/x)=Chser(n-1),Cv(1/y)=Cvser(n-1),C00=C00ser(n-1),eqC+C(x,y)),t,n+1)): fi: end:

map(expand,Cser(6));

KzFJInRHNiIqJClJInhHRiQiIiMhIiIiIiEsJiomSSJ5R0YkIiIiRidGKUYuKiZGLUYuKUYnIiIkRilGLkYuLC4qJClGLUYoRi5GLkYuRi4qKEYoRi5GJkYpRjRGLkYuKiZGKEYuRiZGKUYuKiYpRiciIiVGKUY0Ri5GLiokRjhGKUYuRigsNComRidGLilGLUYxRi5GLiooRihGLkYnRi5GLUYuRi4qKEYxRi5GJ0YpRj1GLkYuKiZGJ0YuRi1GKUYuKigiIidGLkYtRi5GJ0YpRi4qKEYxRi5GMEYpRj1GLkYuKihGQkYuRi1GLkYwRilGLiomKUYnIiImRilGPUYuRi4qKEYoRi5GRkYpRi1GLkYuRjEsRComRiZGLilGLUY5Ri5GLiooRjFGLkYmRi5GNEYuRi4qJkY5Ri5GS0YuRi4qJkYxRi5GJkYuRi4qJiIjN0YuRjRGLkYuKihGQkYuRiZGKUZLRi5GLiomRiZGLkY0RilGLiIiKkYuKigiIz1GLkYmRilGNEYuRi4qKEY5Ri5GOEYpRktGLkYuKiRGNEYpRi4qJkZQRi5GJkYpRi4qKEZQRi5GOEYpRjRGLkYuKiYpRidGQkYpRktGLkYuKiYiIilGLkY4RilGLiooRjFGLkZlbkYpRjRGLkYuKiZGKEYuRmVuRilGLkY5LE4qJkYwRi4pRi1GR0YuRi4qKEY5Ri5GMEYuRj1GLkYuKihGR0YuRidGLkZcb0YuRi4qKEZCRi5GMEYuRi1GLkYuKigiIz9GLkYnRi5GPUYuRi4qKCIjNUYuRidGKUZcb0YuRi4qKEY5Ri5GMEYuRi1GKUYuKigiI0ZGLkYnRi5GLUYuRi4qKCIjU0YuRidGKUY9Ri5GLiooRmNvRi5GMEYpRlxvRi5GLiomRjBGLkY9RilGLiooIiM5Ri5GJ0YuRi1GKUYuKigiI15GLkYtRi5GJ0YpRi4qKEZob0YuRjBGKUY9Ri5GLiooRkdGLkZGRilGXG9GLkYuKihGKEYuRidGLkY9RilGLiooIiNdRi5GLUYuRjBGKUYuKihGYW9GLkZGRilGPUYuRi4qJilGJyIiKEYpRlxvRi5GLiooIiNERi5GRkYpRi1GLkYuKihGOUYuRmZwRilGPUYuRi4qKEZHRi5GZnBGKUYtRi5GLkZHLGBvKiYiJCwiRi5GJkYpRi4qKEZjb0YuRjhGLkY0Ri5GLiooIiNkRi5GJkYuRjRGLkYuKigiJCI9Ri5GJkYpRjRGLkYuKigiJE4iRi5GOEYpRjRGLkYuKihGR0YuRjhGLkZLRi5GLiooIiNJRi5GJkYuRktGLkYuKiZGOEYuKUYtRkJGLkYuKihGQkYuRiZGLkZqcUYuRi4qKCIkKyJGLkYmRilGS0YuRi4qKCIjQEYuRiZGLkY0RilGLiooIiN2Ri5GOEYpRktGLkYuKihGaHFGLkZlbkYpRktGLkYuKigiI2FGLkZlbkYpRjRGLkYuKihGYW9GLkYmRilGanFGLkYuKihGR0YuRjhGLkY0RilGLiooIiM6Ri5GOEYpRmpxRi5GLiomRjhGLkZLRilGLiooRkJGLkZlbkYpRmpxRi5GLiooRjFGLkYmRi5GS0YpRi4qJilGJ0ZnbkYpRmpxRi5GLiomRihGLkZLRilGLiooRkdGLkZdc0YpRktGLkYuKihGU0YuRl1zRilGNEYuRi4qJkZHRi5GXXNGKUYuKiZGXnBGLkYmRi5GLiomIiRRIkYuRjRGLkYuKiZGY29GLkY4Ri5GLiomRmFyRi5GS0YuRi4qJkZhckYuRjhGKUYuKiZGaHFGLkZlbkYpRi4qJiIjO0YuRjRGKUYuKiZGaHJGLkZqcUYuRi4iIyMqRi5GQg==

The orbit sum

n:=6: normal(series(K*(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));

KydJInRHNiIsJCooLCoqJilJInhHRiQiIiMiIiIpSSJ5R0YkRitGLEYsKiRGKUYsISIiKiRGLUYsRjBGLEYsRixGKkYwRi5GMEYwIiIhLUkiT0clKnByb3RlY3RlZEc2I0YsIiIo

expand(%);

KydJInRHNiIsKiomSSJ5R0YkIiIiSSJ4R0YkISIiRigqJkYpRihGJ0YoRioqJkYpRipGJ0YqRioqJkYpRihGJ0YqRigiIiEtSSJPRyUqcHJvdGVjdGVkRzYjRigiIig=

This is minus the orbit sum for quadrant walks

Walks in the quadrant.

eqQ:=-K*Q(x,y)+1-t/y*(x+1/x)*Qp(x)-t/x*(y+1/y)*Qp(y)+t/x/y*Q0;

Expansions

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

Q0ser:=proc(n) subs(x=0,y=0,Qser(n)): end:

Qser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q0=Q0ser(n-1),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);

<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) (i+1)*(j+1)/(1+(i+n)/2)/(1+(j+n)/2)*(n)!^2/((n-i)/2)!/((n+i)/2)!/((n-j)/2)!/((n+j)/2)!: end:

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

NikiIiJGIyIiJSIjRCIkJz4iJWs8IiZDdSI=

i:=2: j:=4: n:=28: coeff(coeff(coeff(convert(Qser(n),polynom),x,i),y,j),t,n)-Qconj(i,j,n);

IiIh

unassign('n'):Qconj('i','j','n');

Qconj(0,0,2*k);

KigpLCZJImtHNiIiIiJGJ0YnIiIjISIiKS1JKmZhY3RvcmlhbEclKnByb3RlY3RlZEc2IywkKiZGKEYnRiVGJ0YnRihGJyktRiw2I0YlIiIlRik=

<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);

LCwqKEkieEc2IiIiIkkieUdGJUYmLUkiUUdGJTYkRiRGJ0YmRiYqKEYnRiZGJCEiIi1GKTYkKiRGJEYsRidGJkYsKihGJEYsRidGLC1GKTYkRi8qJEYnRixGJkYmKihGJEYmRidGLC1GKTYkRiRGM0YmRiwqJiwmRiRGJkYvRixGJiwmRidGJkYzRixGJkYs

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

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);

LCotSSJBRzYiNiRJInhHRiVJInlHRiUiIiIqJiwmRilGKSomIiIjRilJImFHRiVGKSEiIkYpLUkiUUdGJUYmRilGLyooRi5GKSlGJ0YtRi8tRjE2JCokRidGL0YoRilGKSooRi5GKSlGKEYtRi8tRjE2JEYnKiRGKEYvRilGKQ==

Chexpr:=Ah(1/x)+a/x^2*Qp(1/x);

LCYtSSNBaEc2IjYjKiRJInhHRiUhIiIiIiIqKEkiYUdGJUYqKUYoIiIjRiktSSNRcEdGJUYmRipGKg==

Cvexpr:=Av(1/y)+a*Qp(1/y)/y^2;

LCYtSSNBdkc2IjYjKiRJInlHRiUhIiIiIiIqKEkiYUdGJUYqLUkjUXBHRiVGJkYqKUYoIiIjRilGKg==

eqC;

eqA:=collect(subs(C00=A00-(1-2*a)*Q0,C(x,y)=Cexpr,Ch(1/x)=Chexpr,Cv(1/y)=Cvexpr,eqC+(1-2*a)*eqQ-a/x^2*subs(x=1/x,eqQ)-a/y^2*subs(y=1/y,eqQ)),[A,Ah,Av,Q,Qp,Q0,A00],factor);

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

To get rid of Qp

eqA:=collect(subs(a=1/3,eqA),[A,Av,Ah,A00],factor);

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKEYqRjFGLUYxLUkiQUdGJzYkRipGLUYoRigqLEYmRigsJiokRixGKEYoRihGKEYoRipGMUYtRjEtSSNBdkdGJzYjKiRGLUYxRihGMSosRiZGKCwmKiRGKUYoRihGKEYoRihGKkYxRi1GMS1JI0FoR0YnNiMqJEYqRjFGKEYxKipGJkYoRipGMUYtRjFJJEEwMEdGJ0YoRjEqKiNGKCIiJEYoLCgqJkYpRihGLEYoRihGPkYxKiZGK0YoRixGKEYoRihGKUYxRixGMUYo

expand(op(5,eqA));

LCgjIiIiIiIkRiQqJkYjRiQpSSJ5RzYiIiIjISIiRisqJiNGKkYlRiQpSSJ4R0YpRipGK0Yk

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:

map(expand,Aser(4));

Ky1JInRHNiIsKCMiIiIiIiRGJyomRiZGJylJInlHRiQiIiMhIiJGLSomI0YsRihGJylJInhHRiRGLEYtRiciIiEsKiooRiZGJ0YxRidGK0YnRicqJkYxRi1GK0YnRicqKEYmRicpRitGKEYtRjFGJ0YtKihGL0YnKUYxRihGLUYrRidGJ0YnLDoqKEYmRidGMEYnRipGJ0YnKiZGJkYnRjBGJ0YnKiYjIiIlRihGJ0YqRidGJyooRiZGJ0YwRidGKkYtRi1GPkYnKigjIiImRihGJ0YwRi1GKkYnRicqKEYmRidGMEYnKUYrRj9GLUYtRilGLSomRkJGJ0YwRi1GJyooRi9GJylGMUY/Ri1GKkYnRicqJkYmRidGRUYtRi0qJkYvRidGSEYtRidGLCxAKihGJkYnRjlGJ0Y3RidGJyooRi9GJ0Y5RidGK0YnRicqKEZCRidGMUYnRjdGJ0YnKigjIiM1RihGJ0YxRidGK0YnRicqKEYoRidGMUYtRjdGJ0YnKihGL0YnRjlGJ0Y3Ri1GLSomRjFGJ0YrRi1GJyooIiInRidGMUYtRitGJ0YnKigjIiIoRihGJ0Y5Ri1GN0YnRicqKEYmRidGOUYnKUYrRkNGLUYtKihGPkYnRjdGLUYxRidGLSooIyIjOUYoRidGOUYtRitGJ0YnKihGL0YnKUYxRkNGLUY3RidGJyooRi9GJ0YxRidGZW5GLUYtKihGPkYnRltvRi1GK0YnRidGKCxYKiYjIiNLRihGJ0YwRi1GJyomRmhuRidGRUYnRicqJkYvRidGSEYnRicqKCIjO0YnRjBGLUYqRidGJyomRjBGJ0YqRi1GLSooRiZGJ0ZIRidGRUYnRicqKEYsRidGMEYnRkVGJ0YnKiZGSEYnRipGJ0YnKihGKEYnRjBGJ0ZFRi1GLSooIiIqRidGSEYtRipGJ0YnKiZGLEYnRkVGLUYtKiZGVkYnRkhGLUYnKihGL0YnRkhGJ0YqRi1GLSooI0Zlb0YoRidGMEYtRkVGJ0YnKiZGSEYnRkVGLUYtKihGKEYnRkhGLUZFRidGJyooRiZGJ0ZIRicpRitGVkYtRi0qKEYvRicpRjFGVkYtRkVGJ0YnKiZGMEYnRmVwRi1GLSooRixGJ0ZncEYtRipGJ0YnKiZGL0YnRmVwRi1GLSomRj5GJ0ZncEYtRicjIiNKRihGJ0YpRi0qJkZDRidGMEYnRicqJkZpbkYnRipGJ0YnKihGVkYnRjBGJ0YqRidGJ0Y/

Ah is negative in x

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

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

A00ser:=proc(n) coeff(coeff(convert(Aser(n),polynom),y,0),x,0):end:

n:=16: normal(series(subs(A0=A0ser(n),A(x,y)=Aser(n), Ah(1/x)=Ahser(n), Av(1/y)=subs(x=y,Avser(n)), A00=A00ser(n),eqA),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM8

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:

eqA := (t*x^2*y^2+t*x^2+t*y^2-x*y+t)*A(x, y)/(y*x)-t*(y^2+1)*Av(1/y)/(y*x)-t*(x^2+1)*Ah(1/x)/(y*x)-t*A00/(y*x)+(1/3)*(x^2*y^2-x^2+2*y^2)/(x^2*y^2);

LCwqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKC1JIkFHRic2JEYqRi1GKEYqRjFGLUYxRigqLEYmRigsJiokRixGKEYoRihGKEYoLUkjQXZHRic2IyokRi1GMUYoRipGMUYtRjFGMSosRiZGKCwmKiRGKUYoRihGKEYoRigtSSNBaEdGJzYjKiRGKkYxRihGKkYxRi1GMUYxKipGJkYoSSRBMDBHRidGKEYqRjFGLUYxRjEqKiNGKCIiJEYoLCgqJkYpRihGLEYoRihGPkYxKiZGK0YoRixGKEYoRihGKUYxRixGMUYo

expand(3*op(5,%));

LCgiIiJGIyokKUkieUc2IiIiIyEiIkYpKiZGKEYjKUkieEdGJ0YoRilGIw==

We separate in A the contribution of the three quadrants

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

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

Lser:=proc(n) collect(add(1/x*coeff(convert(Aser(n),polynom),x,-i)*x^i,i=1..n+4),t,normal): end:

Lser(3);

LCoqLCMiIiIiIiRGJUkieUc2IkYlLCYqJClGJyIiI0YlRiVGLEYlRiUsKComRixGJSlJInhHRigiIiVGJUYlKiYiIihGJSlGMEYsRiVGJSIiKkYlRiUpSSJ0R0YoRiZGJUYlKiYsKiooIyIiJkYmRiVGMEYlRitGJUYlKiZGO0YlRjBGJUYlKigjRixGJkYlKUYwRiZGJUYrRiVGJSomRj9GJUZARiVGJUYlKUY3RixGJUYlKiYsJkYnRiUqKEY/RiVGNEYlRidGJUYlRiVGN0YlRiUqJkY/RiVGMEYlRiU=

Bser:=proc(n) collect(add(1/y*coeff(convert(Aser(n),polynom),y,-i)*y^i,i=1..n+4),t,normal): end:

Bser(6);

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

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

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+4),j=0..n+4),t): end:

Pser(3);

LCojIiIiIiIkRiQqJiwqKigjIiM1RiVGJEkieEc2IkYkSSJ5R0YsRiRGJCooIyIiI0YlRiQpRitGJUYkRi1GJEYkKigjIiImRiVGJEYrRiQpRi1GJUYkRiQqKEYjRiRGMUYkRjVGJEYkRiQpSSJ0R0YsRiVGJEYkKiYsKiMiIiVGJUYkKiZGI0YkKUYrRjBGJEYkKiZGO0YkKUYtRjBGJEYkKihGI0YkRj5GJEZARiRGJEYkKUY4RjBGJEYkKipGI0YkRitGJEYtRiRGOEYkRiQ=

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

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

LCoqKEkieEc2IiIiIkkieUdGJUYmLUkiQUdGJTYkRiRGJ0YmRiYqKEYnRiZGJCEiIi1GKTYkKiRGJEYsRidGJkYsKihGJEYmRidGLC1GKTYkRiQqJEYnRixGJkYsKihGJEYsRidGLC1GKTYkRi9GM0YmRiY=

We plug in the orbit equation the expression of A

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

LCgtSSJQRzYiNiRJInhHRiVJInlHRiUiIiIqJkYnISIiLUkiTEdGJTYkKiRGJ0YrRihGKUYpKiZGKEYrLUkiQkdGJTYkRicqJEYoRitGKUYp

eqo:=expand(subs(A(x,y)=Aexpr,A(1/x,y)=subs(x=1/x,Aexpr), A(x,1/y)=subs(y=1/y,Aexpr),A(1/x,1/y)=subs(x=1/x,y=1/y,Aexpr),eqorbitA));

LDoqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYtSSJMR0YlNiQqJEYkISIiRidGJkYmKiZGJEYmLUkiQkdGJTYkRiQqJEYnRjBGJkYmKihGJEYwRidGJi1GKUYuRiZGMComRidGJi1GLUYqRiZGMComRiRGMC1GMzYkRi9GNUYmRjAqKEYkRiZGJ0YwLUYpRjRGJkYwKiZGJ0YwLUYtRjxGJkYwKiZGJEYmLUYzRipGJkYwKihGJEYwRidGMC1GKUY8RiZGJiomRidGMC1GLUY0RiZGJiomRiRGMC1GM0YuRiZGJg==

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

IiIh

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=

Let us check this identity

n:=12: normal(series(subs(P(x,y)=Pser(n), L(x,y)=Lser(n),L(0,y)=subs(x=0,Lser(n)),B(x,y)=Bser(n),B(x,0)=subs(y=0,Bser(n)),eqospxy),t,n+1));

IiIh

Express P in terms of L and B

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

LCoqJkkieEc2IiEiIi1JIkxHRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1JIkJHRiVGKUYrRisqJkYqRiYtRjI2JEYkRi9GK0Ym

Checking the equivalence with the OS

AexprLB:=subs(P(x,y)=Pexpr,Aexpr);

LC4qJkkieEc2IiEiIi1JIkxHRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1JIkJHRiVGKUYrRisqJkYqRiYtRjI2JEYkRi9GK0YmKiZGJEYmLUYoNiQqJEYkRiZGKkYrRisqJkYqRiYtRjI2JEYkKiRGKkYmRitGKw==

normal(subs(A(x,y)=AexprLB,A(1/x,y)=subs(x=1/x,AexprLB), A(x,1/y)=subs(y=1/y,AexprLB), A(1/x,1/y)=subs(x=1/x,y=1/y, AexprLB),eqorbitA));

IiIh

We go back to the equation defining A. We need to express A00=P00 in terms of L and B

limit(convert(series(Pexpr,x,2),polynom),y=0);

LCYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIjYjSSJMR0YqNiQiIiFGMEYsLS0mRiY2IyIiIzYjSSJCR0YqRi9GLA==

eqLB:=collect(subs(A(x,y)=AexprLB, Ah(1/x)=1/x*L(1/x,0), Av(1/y)=1/y*B(0,1/y),A00=L10+B01,eqA),[B,L]);

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

Checking

indets(eqLB);

PC9JJEIwMUc2IkkkTDEwR0YkSSJ0R0YkSSJ4R0YkSSJ5R0YkLUkiQkdGJDYkIiIhKiRGKCEiIi1GKjYkRidGLC1GKjYkRidGKC1GKjYkRidGLS1JIkxHRiQ2JEYsRigtRjZGMi1GNjYkKiRGJ0YuRiwtRjY2JEY7Rig=

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), L10=coeff(coeff(Lser(n),x,1),y,0), B01=coeff(coeff(Bser(n),x,0),y,1),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 from eqLB. Recall that B00=0=L00

subs(x=0,y=0,Bser(n));subs(x=0,y=0,Lser(n));

IiIh

map(factor,eqLB);

The series in x

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

KylJInhHNiIjIiIjIiIkISIjLCQqKEkidEdGJCIiIiwuKiYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsNiNJIkxHRiQ2JCIiIUkieUdGJEYsKUY7RidGLCEiIiomLUkiQkdGJEY5RiwpRjtGJkYsRj0qJkY7RixJJEIwMUdGJEYsRiwqJkY7RixJJEwxMEdGJEYsRiwqJkYvRixGO0YsRj1GP0Y9RixGQUY9Rj1GPS1JIk9HRjRGNkY6

The series in 1/x

op(7,eqLB)+op(8,eqLB);

LCYqKiwsKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRilGKEYoKiZGJkYoRixGKEYoKiZGKkYoRi1GKCEiIkYmRihGKEYpRjFGLUYxLUkiTEdGJzYkKiRGKkYxRi1GKEYoKixGJkYoLCYqJEYpRihGKEYoRihGKEYpRjFGLUYxLUYzNiRGNSIiIUYoRjE=

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

KytJInhHNiIqKkkidEdGJCIiIiwmKiQpSSJ5R0YkIiIjRidGJ0YnRidGJy1JIkxHRiQ2JCIiIUYrRidGKyEiIkYwKiYsKiooLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJzYjRi5GL0YnRiZGJ0YqRidGJyomRiZGJ0Y1RidGJyomRiZGJy1GNjYkRjBGMEYnRjEqJkYrRidGLUYnRjFGJ0YrRjFGJywkKigjRidGLEYnLC4qKC0tJkY4NiRGJ0YnRj1GL0YnRiZGJ0YqRidGMSoqRixGJ0YtRidGJkYnRipGJ0YxKiZGJkYnLUZJRkFGJ0YnKiZGJkYnRkhGJ0YxKihGLEYnRjVGJ0YrRidGJyooRixGJ0YmRidGLUYnRjFGJ0YrRjFGMUYsLUkiT0dGOkY8IiIk

eqLBxneg:=convert(%%%,polynom)+op(7,eqLB)+op(8,eqLB)-coeff(%,x,0);

LCwqJiMiIiMiIiQiIiIpSSJ4RzYiRiUhIiJGJyoqSSJ0R0YqRicsLiomLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YqNiNGJzYjSSJMR0YqNiQiIiFJInlHRipGJylGPEYmRidGKyomLUkiQkdGKkY6RicpRjxGJUYnRisqJkY8RidJJEIwMUdGKkYnRicqJkY8RidJJEwxMEdGKkYnRicqJkYwRidGPEYnRitGP0YrRidGQUYrRilGK0YrKiosLCooRi1GJ0YoRidGQUYnRicqJkYtRidGKEYnRicqJkYtRidGQUYnRicqJkYpRidGPEYnRitGLUYnRidGKEYrRjxGKy1GOTYkKiRGKUYrRjxGJ0YnKixGLUYnLCYqJEYoRidGJ0YnRidGJ0YoRitGPEYrLUY5NiRGT0Y7RidGKyoqRi1GJywmKiRGQUYnRidGJ0YnRictRjlGOkYnRjxGK0Yr

Checking that what remains is non-negative in x

normal(subs(B(0,0)=0,series(normal(eqLB-eqLBxneg),x,3)));

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

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

normal(subs(L(0,0)=0,series(subs(x=1/x,eqLBxneg),x,3)));

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

This gives us an expression of 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

D1Lsol:=solve(subs((D[1](L))(0, 0)=L10,coeff(%,x)),(D[1](L))(0, y));

LCQqLCMiIiIiIiNGJSwsKigtSSJCRzYiNiQiIiFJInlHRitGJUkidEdGK0YlKUYuRiZGJSEiIiooSSRCMDFHRitGJUYvRiVGLkYlRiUqKkYmRiVJJEwxMEdGK0YlRi9GJUYuRiVGJSomLUkiTEdGK0YsRiVGMEYlRiUqJkYpRiVGL0YlRjFGJUYuRjFGL0YxLCYqJEYwRiVGJUYlRiVGMUYl

eqLBxneg:=collect(subs((D[1](L))(0, y)=D1Lsol,eqLBxneg),[L,B,B01],factor);

LC4qLCMiIiIiIiNGJSwoKipGJkYlSSJ0RzYiRiVJInhHRipGJSlJInlHRipGJkYlRiUqKEYmRiVGKUYlRitGJUYlRi0hIiJGJUYtRi9GK0YvLUkiTEdGKjYkIiIhRi1GJUYvKixGKUYlLCYqJClGK0YmRiVGJUYlRiVGJUY3Ri9GLUYvLUYxNiQqJEYrRi9GM0YlRi8qKiwsKihGKUYlRjdGJUYsRiVGJSomRilGJUY3RiVGJSomRilGJUYsRiVGJSomRitGJUYtRiVGL0YpRiVGJUY3Ri9GLUYvLUYxNiRGOkYtRiVGJSouRiRGJUYpRiUsJiokRixGJUYlRiVGJUYlLUkiQkdGKkYyRiVGK0YvRixGL0YlKixGJEYlRilGJUYrRi9GLUYvSSRCMDFHRipGJUYvKiYjRiYiIiRGJUY3Ri9GJQ==

eqLB4:=map(factor,subs(x=1/x,%));

LC4qKiMiIiIiIiNGJSwoKihGJkYlSSJ0RzYiRiUpSSJ5R0YqRiZGJUYlKiZJInhHRipGJUYsRiUhIiIqJkYmRiVGKUYlRiVGJS1JIkxHRio2JCIiIUYsRiVGLEYvRi8qKkYpRiUsJiokKUYuRiZGJUYlRiVGJUYlLUYyNiRGLkY0RiVGLEYvRi8qKCwsKihGKUYlRjhGJUYrRiVGJSomRilGJUY4RiVGJSomRilGJUYrRiVGJUYtRi9GKUYlRiUtRjI2JEYuRixGJUYsRi9GJSouRiRGJUYpRiUsJiokRitGJUYlRiVGJUYlLUkiQkdGKkYzRiVGLkYlRitGL0YlKixGJEYlRilGJUYuRiVGLEYvSSRCMDFHRipGJUYvKiYjRiYiIiRGJUY4RiVGJQ==

indets(eqLB4);

PCpJJEIwMUc2IkkidEdGJEkieEdGJEkieUdGJC1JIkJHRiQ2JCIiIUYnLUkiTEdGJEYqLUYtNiRGJkYrLUYtNiRGJkYn

n:=10: normal(series(subs( B(0,y)=subs(x=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), L10=coeff(coeff(Lser(n),x,1),y,0), B01=coeff(coeff(Bser(n),x,0),y,1),eqLB4),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

We have obtained an equation for L

eqL:=collect(eqLB4/x*2,[L,B,B01],factor);

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnRitGLkYtRi4tSSJMR0YpNiQiIiFGK0YnRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJ0YtRi5GK0YuLUYxNiRGLUYzRidGLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRidGLUYuRitGLi1GMTYkRi1GK0YnRicqKkYoRicsJiokRipGJ0YnRidGJ0YnRipGLi1JIkJHRilGMkYnRicqKEYoRidGK0YuSSRCMDFHRilGJ0YuKiYjIiIlIiIkRidGLUYnRic=

eqLbis:=collect(subs(L(x,y)=Lbis(x,y)+1/2*L(0,y),eqL),[Lbis,L,B,B01],factor);

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiVGK0YxRi1GMS1JJUxiaXNHRik2JEYrRi1GJUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJUYrRjFGLUYxLUkiTEdGKTYkIiIhRi1GJUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiVGK0YxRi1GMS1GOzYkRitGPUYlRjEqKkYoRiVGNkYlRixGMS1JIkJHRilGPEYlRiUqKEYoRiVGLUYxSSRCMDFHRilGJUYxKiYjIiIlIiIkRiVGK0YlRiU=

We now get back to eqLB and extract the y-negative part as we did above for the x-negative part.

eqLB;

The series in y

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

KylJInlHNiIjISIiIiIkISIjLCQqKEkidEdGJCIiIiwuKiYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIzYjSSJCR0YkNiRJInhHRiQiIiFGLClGO0YnRixGJiomLUkiTEdGJEY6RiwpRjtGN0YsRiYqJkY7RixJJEIwMUdGJEYsRiwqJkY7RixJJEwxMEdGJEYsRiwqJkYvRixGO0YsRiZGP0YmRixGQUYmRiZGJi1JIk9HRjQ2I0YsRjw=

The series in 1/y

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

LCYqLEkidEc2IiIiIiwmKiQpSSJ5R0YlIiIjRiZGJkYmRiZGJkkieEdGJSEiIkYpRi0tSSJCR0YlNiQiIiEqJEYqRi1GJkYtKiosLCooRiRGJilGLEYrRiZGKUYmRiYqJkYkRiZGNkYmRiYqJkYkRiZGKUYmRiYqJkYsRiZGKkYmRi1GJEYmRiZGLEYtRilGLS1GLzYkRixGMkYmRiY=

normal(subs(B(0,0)=0,series(subs(y=1/y,%),y,3)));

KytJInlHNiIqKkkidEdGJCIiIiwmKiQpSSJ4R0YkIiIjRidGJ0YnRidGJy1JIkJHRiQ2JEYrIiIhRidGKyEiIkYwKiYsKiooLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLDYjRi5GL0YnRiZGJ0YqRidGJyomRiZGJ0Y1RidGJyomRiZGJy1GNjYkRjBGMEYnRjEqJkYrRidGLUYnRjFGJ0YrRjFGJywkKigjRidGLEYnLC4qKC0tJkY4NiRGLEYsRj1GL0YnRiZGJ0YqRidGJyoqRixGJ0YtRidGJkYnRipGJ0YnKiZGJkYnRkhGJ0YnKihGLEYnRjVGJ0YrRidGMSomRiZGJy1GSUZBRidGMSooRixGJ0YmRidGLUYnRidGJ0YrRjFGJ0YsLUkiT0dGOjYjRiciIiQ=

eqLByneg:=convert(%%%,polynom)+op(1,eqLB)+op(4,eqLB)-coeff(%,y,0);

LCwqJiMiIiIiIiRGJSlJInlHNiIiIiMhIiJGKyoqSSJ0R0YpRiUsLiomLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGKjYjSSJCR0YpNiRJInhHRikiIiFGJSlGO0YmRiVGKyomLUkiTEdGKUY6RiUpRjtGKkYlRisqJkY7RiVJJEIwMUdGKUYlRiUqJkY7RiVJJEwxMEdGKUYlRiUqJkYwRiVGO0YlRitGP0YrRiVGQUYrRihGK0YrKixGLUYlLCYqJEYnRiVGJUYlRiVGJUY7RitGJ0YrLUY5NiRGPCokRihGK0YlRisqKiwsKihGLUYlRkFGJUYnRiVGJSomRi1GJUZBRiVGJSomRi1GJUYnRiVGJSomRjtGJUYoRiVGK0YtRiVGJUY7RitGJ0YrLUY5NiRGO0ZMRiVGJSoqRi1GJSwmKiRGQUYlRiVGJUYlRiUtRjlGOkYlRjtGK0Yr

Checking that what remains is non-negative in y

normal(subs(L(0,0)=0,series(normal(eqLB-eqLByneg),y,3)));

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

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

normal(subs(B(0,0)=0,series(subs(y=1/y,eqLByneg),y,3)));

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

This gives us an expression of 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

D2Bsol:=solve(subs((D[2](B))(0, 0)=B01,coeff(%,y)),(D[2](B))(x,0));

LCQqLCMiIiIiIiNGJSwsKigtSSJMRzYiNiRJInhHRisiIiFGJUkidEdGK0YlKUYtRiZGJSEiIioqRiZGJUYvRiVGLUYlSSRCMDFHRitGJUYlKihJJEwxMEdGK0YlRi9GJUYtRiVGJSomLUkiQkdGK0YsRiVGMEYlRiUqJkYpRiVGL0YlRjFGJUYtRjFGL0YxLCYqJEYwRiVGJUYlRiVGMUYl

eqLByneg:=collect(subs((D[2](B))(x,0)=D2Bsol,eqLByneg),[L,B,L10],factor);

LC4qLiMiIiIiIiNGJUkidEc2IkYlLCYqJClJInhHRihGJkYlRiVGJUYlRiVGKyEiIkkieUdGKEYtLUkiTEdGKDYkRiwiIiFGJUYlKixGJ0YlLCYqJClGLkYmRiVGJUYlRiVGJUYsRi1GNkYtLUkiQkdGKDYkRjIqJEYuRi1GJUYtKixGJEYlLCgqKkYmRiVGJ0YlRitGJUYuRiVGJSooRiZGJUYnRiVGLkYlRiVGLEYtRiVGLEYtRi5GLS1GOEYxRiVGLSoqLCwqKEYnRiVGK0YlRjZGJUYlKiZGJ0YlRitGJUYlKiZGJ0YlRjZGJUYlKiZGLEYlRi5GJUYtRidGJUYlRixGLUY2Ri0tRjg2JEYsRjpGJUYlKixGJEYlRidGJUYsRi1GLkYtSSRMMTBHRihGJUYtKiYjRiUiIiRGJUY2Ri1GLQ==

eqLB5:=map(factor,subs(y=1/y,%));

LC4qLiMiIiIiIiNGJUkidEc2IkYlLCYqJClJInhHRihGJkYlRiVGJUYlRiVGKyEiIkkieUdGKEYlLUkiTEdGKDYkRiwiIiFGJUYlKipGJ0YlLCYqJClGLkYmRiVGJUYlRiVGJS1JIkJHRig2JEYyRi5GJUYsRi1GLSoqRiRGJSwoKihGJkYlRidGJUYrRiVGJSomRixGJUYuRiVGLSomRiZGJUYnRiVGJUYlLUY4RjFGJUYsRi1GLSooLCwqKEYnRiVGK0YlRjZGJUYlKiZGJ0YlRitGJUYlKiZGJ0YlRjZGJUYlRj1GLUYnRiVGJS1GODYkRixGLkYlRixGLUYlKixGJEYlRidGJUYsRi1GLkYlSSRMMTBHRihGJUYtKiYjRiUiIiRGJUY2RiVGLQ==

indets(eqLB5);

PCpJJEwxMEc2IkkidEdGJEkieEdGJEkieUdGJC1JIkJHRiQ2JCIiIUYnLUYpNiRGJkYrLUYpNiRGJkYnLUkiTEdGJEYt

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),L10=coeff(coeff(Lser(n),x,1),y,0), B01=coeff(coeff(Bser(n),x,0),y,1), eqLB5),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

We have obtained an equation for B

eqB:=collect(eqLB5*2/y,[L,B,L10],factor);

LC4qKkkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJkYpISIiLUkiTEdGJTYkRioiIiFGJkYmKi5GK0YmRiRGJiwmKiQpSSJ5R0YlRitGJkYmRiZGJkYmRipGLEY1RiwtSSJCR0YlNiRGMEY1RiZGLCoqLCgqKEYrRiZGJEYmRilGJkYmKiZGKkYmRjVGJkYsKiZGK0YmRiRGJkYmRiZGKkYsRjVGLC1GN0YvRiZGLCosRitGJiwsKihGJEYmRilGJkY0RiZGJiomRiRGJkYpRiZGJiomRiRGJkY0RiZGJkY8RixGJEYmRiZGKkYsRjVGLC1GNzYkRipGNUYmRiYqKEYkRiZGKkYsSSRMMTBHRiVGJkYsKiYjRisiIiRGJkY1RiZGLA==

eqBbis:=collect(subs(B(x,y)=Bbis(x,y)+1/2*B(x,0),eqB),[Bbis,L,B,L10],factor);

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiVGK0YxRi1GMS1JJUJiaXNHRik2JEYrRi1GJUYlKipGKEYlLCYqJEYqRiVGJUYlRiVGJUYqRjEtSSJMR0YpNiRGKyIiIUYlRiUqLkYkRiVGKEYlLCYqJEYsRiVGJUYlRiVGJUYrRjFGLUYxLUkiQkdGKTYkRjtGLUYlRjEqMEYoRiUsJkYtRiVGJUYxRiUsJkYtRiVGJUYlRiVGNkYlRitGMUYtRjEtRkBGOkYlRiUqKEYoRiVGK0YxSSRMMTBHRilGJUYxKiYjRiQiIiRGJUYtRiVGMQ==

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

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

Ly1JIkJHNiI2JEkieEdGJUkieUdGJSwkKiwjIiIiIiIjRiwsLCoqSSJ0R0YlRiwsJiokKUYnRi1GLEYsRixGLEYsRjMhIiItSSJMR0YlNiRGJyIiIUYsRjQqLkYtRixGMEYsLCYqJClGKEYtRixGLEYsRixGLEYnRjRGKEY0LUYkNiRGOEYoRixGLCoqLCgqKEYtRixGMEYsRjNGLEYsKiZGJ0YsRihGLEY0KiZGLUYsRjBGLEYsRixGJ0Y0RihGNC1GJEY3RixGLCooRjBGLEYnRjRJJEwxMEdGJUYsRiwqJiNGLSIiJEYsRihGLEYsRiwsLCooRjBGLEYzRixGPEYsRiwqJkYwRixGM0YsRiwqJkYwRixGPEYsRixGQkY0RjBGLEY0RidGLEYoRixGLA==

Ly1JIkxHNiI2JEkieEdGJUkieUdGJSwkKiwjIiIiIiIjRiwsLCoqLCgqKEYtRixJInRHRiVGLClGKEYtRixGLComRidGLEYoRiwhIiIqJkYtRixGMkYsRixGLEYoRjVGJ0Y1LUYkNiQiIiFGKEYsRiwqLkYtRixGMkYsLCYqJClGJ0YtRixGLEYsRixGLEYnRjVGKEY1LUYkNiRGJ0Y5RixGLCoqRjJGLCwmKiRGM0YsRixGLEYsRixGM0Y1LUkiQkdGJUY4RixGNSooRjJGLEYoRjVJJEIwMUdGJUYsRiwqJiMiIiUiIiRGLEYnRixGNUYsLCwqKEYyRixGPUYsRjNGLEYsKiZGMkYsRj1GLEYsKiZGMkYsRjNGLEYsRjRGNUYyRixGNUYnRixGKEYsRiw=

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

IiIh

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

LCoqJi1JIkxHNiI2JEkieEdGJkkieUdGJiIiIkYoISIiRioqJi1GJTYkIiIhRilGKkYoRitGKyomLUkiQkdGJkYnRipGKUYrRioqJi1GMjYkRihGL0YqRilGK0Yr

eqL:=-(2*t*y^2-x*y+2*t)*L(0, y)/(y*x)-2*t*(x^2+1)*L(x, 0)/(x*y)+(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*L(x, y)/(x*y)+t*(y^2+1)*B(0, y)/y^2-t*B01/y+(4/3)*x; eqB:=t*(x^2+1)*L(x, 0)/x^2-2*t*(y^2+1)*B(0, y)/(x*y)-(2*t*x^2-x*y+2*t)*B(x, 0)/(x*y)+(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*B(x, y)/(x*y)-t*L10/x-(2/3)*y;

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnLUkiTEdGKTYkIiIhRitGJ0YtRi5GK0YuRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJy1GMTYkRi1GM0YnRi1GLkYrRi5GLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRictRjE2JEYtRitGJ0YtRi5GK0YuRicqKkYoRicsJiokRipGJ0YnRidGJ0YnLUkiQkdGKUYyRidGKkYuRicqKEYoRidJJEIwMUdGKUYnRitGLkYuKiYjIiIlIiIkRidGLUYnRic=

LC4qKkkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJi1JIkxHRiU2JEYqIiIhRiZGKSEiIkYmKi5GK0YmRiRGJiwmKiQpSSJ5R0YlRitGJkYmRiZGJkYmLUkiQkdGJTYkRi9GNUYmRipGMEY1RjBGMCoqLCgqKEYrRiZGJEYmRilGJkYmKiZGKkYmRjVGJkYwKiZGK0YmRiRGJkYmRiYtRjdGLkYmRipGMEY1RjBGMCosRitGJiwsKihGJEYmRilGJkY0RiZGJiomRiRGJkYpRiZGJiomRiRGJkY0RiZGJkY8RjBGJEYmRiYtRjc2JEYqRjVGJkYqRjBGNUYwRiYqKEYkRiZJJEwxMEdGJUYmRipGMEYwKiYjRisiIiRGJkY1RiZGMA==

eqLbis:=(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*Lbis(x, y)/(x*y)+t*(y^2+1)*(x-1)*(x+1)*L(0, y)/(x*y)-2*t*(x^2+1)*L(x, 0)/(x*y)+t*(y^2+1)*B(0, y)/y^2-t*B01/y+(4/3)*x; eqBbis:=(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*Bbis(x, y)/(x*y)+t*(x^2+1)*L(x, 0)/x^2-2*t*(y^2+1)*B(0, y)/(x*y)+t*(y-1)*(y+1)*(x^2+1)*B(x, 0)/(x*y)-t*L10/x-(2/3)*y;

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiUtSSVMYmlzR0YpNiRGK0YtRiVGK0YxRi1GMUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJS1JIkxHRik2JCIiIUYtRiVGK0YxRi1GMUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiUtRjs2JEYrRj1GJUYrRjFGLUYxRjEqKkYoRiVGNkYlLUkiQkdGKUY8RiVGLEYxRiUqKEYoRiVJJEIwMUdGKUYlRi1GMUYxKiYjIiIlIiIkRiVGK0YlRiU=

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiUtSSVCYmlzR0YpNiRGK0YtRiVGK0YxRi1GMUYlKipGKEYlLCYqJEYqRiVGJUYlRiVGJS1JIkxHRik2JEYrIiIhRiVGKkYxRiUqLkYkRiVGKEYlLCYqJEYsRiVGJUYlRiVGJS1JIkJHRik2JEY7Ri1GJUYrRjFGLUYxRjEqMEYoRiUsJkYtRiVGJUYxRiUsJkYtRiVGJUYlRiVGNkYlLUZARjpGJUYrRjFGLUYxRiUqKEYoRiVJJEwxMEdGKUYlRitGMUYxKiYjRiQiIiRGJUYtRiVGMQ==

let us check

n:=15: normal(series(subs(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)),L10=coeff(coeff(Lser(n),x,1),y,0), B01=coeff(coeff(Bser(n),x,0),y,1), eqL),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

n:=15: normal(series(subs(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)),L10=coeff(coeff(Lser(n),x,1),y,0), B01=coeff(coeff(Bser(n),x,0),y,1), eqB),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

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

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

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

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),L10=M10-B01, %),[M,L,B,M10],factor);

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnRitGLkYtRi4tSSJNR0YpNiQiIiFGK0YnRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJ0YtRi5GK0YuLUYxNiRGLUYzRidGLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRidGLUYuRitGLi1GMTYkRi1GK0YnRicqKkYoRicsJiokRipGJ0YnRidGJ0YnRipGLi1GMTYkRitGM0YnRicqKEYoRidGK0YuSSRNMTBHRilGJ0YuKiYjRiYiIiRGJ0YtRidGJw==

subs(t=0,eqM);

LCgtSSJNRzYiNiQiIiFJInlHRiUiIiIqJiIiI0YpLUYkNiRJInhHRiVGKEYpISIiKiYjRisiIiRGKUYuRilGKQ==

eqMbis:=collect(subs(M(x,y)=Mbis(x,y)+1/2*M(0,y),eqM),[Mbis,M,M10],factor);

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiVGK0YxRi1GMS1JJU1iaXNHRik2JEYrRi1GJUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJUYrRjFGLUYxLUkiTUdGKTYkIiIhRi1GJUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiVGK0YxRi1GMS1GOzYkRitGPUYlRjEqKkYoRiVGNkYlRixGMS1GOzYkRi1GPUYlRiUqKEYoRiVGLUYxSSRNMTBHRilGJUYxKiYjRiQiIiRGJUYrRiVGJQ==

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

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

eqN:=collect(subs(L(x,y)=N(x,y)+B(y,x),L10=N10+B01, 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), %),[N,L,B,N10],factor);

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnRitGLkYtRi4tSSJOR0YpNiQiIiFGK0YnRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJ0YtRi5GK0YuLUYxNiRGLUYzRidGLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRidGLUYuRitGLi1GMTYkRi1GK0YnRicqKkYoRicsJiokRipGJ0YnRidGJ0YnRipGLi1GMTYkRitGM0YnRi4qKEYoRidGK0YuSSROMTBHRilGJ0YnKiZGJkYnRi1GJ0Yn

eqNbis:=collect(subs(N(x,y)=Nbis(x,y)+1/2*N(0,y),eqN),[Nbis,N,N10],factor);

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiVGK0YxRi1GMS1JJU5iaXNHRik2JEYrRi1GJUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJUYrRjFGLUYxLUkiTkdGKTYkIiIhRi1GJUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiVGK0YxRi1GMS1GOzYkRitGPUYlRjEqKkYoRiVGNkYlRixGMS1GOzYkRi1GPUYlRjEqKEYoRiVGLUYxSSROMTBHRilGJUYlKiZGJEYlRitGJUYl

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);

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

The equation for M is the one we solved when counting walks starting at (0,0) !

eqM:=-(2*t*y^2-x*y+2*t)*M(0, y)/(y*x)-2*t*(x^2+1)*M(x, 0)/(x*y)+(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*M(x, y)/(x*y)+t*(y^2+1)*M(y, 0)/y^2-t*M10/y+(2/3)*x;

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnLUkiTUdGKTYkIiIhRitGJ0YtRi5GK0YuRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJy1GMTYkRi1GM0YnRi1GLkYrRi5GLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRictRjE2JEYtRitGJ0YtRi5GK0YuRicqKkYoRicsJiokRipGJ0YnRidGJ0YnLUYxNiRGK0YzRidGKkYuRicqKEYoRidGK0YuSSRNMTBHRilGJ0YuKiYjRiYiIiRGJ0YtRidGJw==

eqMbis:=(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*Mbis(x, y)/(x*y)+t*(y^2+1)*(x-1)*(x+1)*M(0, y)/(x*y)-2*t*(x^2+1)*M(x, 0)/(x*y)+t*(y^2+1)*M(y, 0)/y^2-t*M10/y+(2/3)*x;

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiUtSSVNYmlzR0YpNiRGK0YtRiVGK0YxRi1GMUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJS1JIk1HRik2JCIiIUYtRiVGK0YxRi1GMUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiUtRjs2JEYrRj1GJUYrRjFGLUYxRjEqKkYoRiVGNkYlLUY7NiRGLUY9RiVGLEYxRiUqKEYoRiVGLUYxSSRNMTBHRilGJUYxKiYjRiQiIiRGJUYrRiVGJQ==

So we just have to solve the equation for N

eqN:=-(2*t*y^2-x*y+2*t)*N(0, y)/(y*x)-2*t*(x^2+1)*N(x, 0)/(x*y)+(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*N(x, y)/(x*y)-t*(y^2+1)*N(y, 0)/y^2+t*N10/y+2*x;

LC4qKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnLUkiTkdGKTYkIiIhRitGJ0YtRi5GK0YuRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJy1GMTYkRi1GM0YnRi1GLkYrRi5GLiosRiZGJywsKihGKEYnRjdGJ0YqRidGJyomRihGJ0Y3RidGJyomRihGJ0YqRidGJ0YsRi5GKEYnRictRjE2JEYtRitGJ0YtRi5GK0YuRicqKkYoRicsJiokRipGJ0YnRidGJ0YnLUYxNiRGK0YzRidGKkYuRi4qKEYoRidJJE4xMEdGKUYnRitGLkYnKiZGJkYnRi1GJ0Yn

eqNbis:=(2*(t*x^2*y^2+t*x^2+t*y^2-x*y+t))*Nbis(x, y)/(x*y)+t*(y^2+1)*(x-1)*(x+1)*N(0, y)/(x*y)-2*t*(x^2+1)*N(x, 0)/(x*y)-t*(y^2+1)*N(y, 0)/y^2+t*N10/y+2*x;

LC4qLCIiIyIiIiwsKihJInRHNiJGJSlJInhHRilGJEYlKUkieUdGKUYkRiVGJSomRihGJUYqRiVGJSomRihGJUYsRiVGJSomRitGJUYtRiUhIiJGKEYlRiUtSSVOYmlzR0YpNiRGK0YtRiVGK0YxRi1GMUYlKjBGKEYlLCYqJEYsRiVGJUYlRiVGJSwmRitGJUYlRjFGJSwmRitGJUYlRiVGJS1JIk5HRik2JCIiIUYtRiVGK0YxRi1GMUYlKi5GJEYlRihGJSwmKiRGKkYlRiVGJUYlRiUtRjs2JEYrRj1GJUYrRjFGLUYxRjEqKkYoRiVGNkYlLUY7NiRGLUY9RiVGLEYxRjEqKEYoRiVJJE4xMEdGKUYlRi1GMUYlKiZGJEYlRitGJUYl

n:=15: normal(series(subs(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)), M10=coeff(coeff(Mser(n),x,1),y,0),eqM),t,n+1)); normal(series(subs(N10=coeff(coeff(Nser(n),x,1),y,0),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)]);

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

LCQqKiMiIiIiIiNGJSwmSSJ4RzYiISIiKiQpLCQqJiwoKihGJkYlSSJ0R0YpRiUpRihGJkYlRiUqJkYmRiVGMUYlRiVGKEYlRiUsKEYwRiVGM0YlRihGKkYlRipGJEYlRiVGJUYxRiosJiokRjJGJUYlRiVGJUYqRio=

LCQqKiMiIiIiIiNGJSwmSSJ4RzYiRiUqJCksJComLCgqKEYmRiVJInRHRilGJSlGKEYmRiVGJSomRiZGJUYwRiVGJUYoRiVGJSwoRi9GJUYyRiVGKCEiIkYlRjRGJEYlRiVGJUYwRjQsJiokRjFGJUYlRiVGJUY0RiU=

Delta:=1-4*t^2*(x+1/x)^2;

LCYiIiJGIyooIiIlRiMpSSJ0RzYiIiIjRiMpLCZJInhHRihGIyokRiwhIiJGI0YpRiNGLg==

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

Y0ser(4);

KylJInRHNiIqJiwmKiQpSSJ4R0YkIiIjIiIiRitGK0YrRitGKSEiIkYrKiYpRikiIiRGLClGJkYvRitGLy1JIk9HJSpwcm90ZWN0ZWRHNiNGKyIiJg==

factor(expand(Y0sol*Y1sol));

IiIi

<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>

Cancel the kernel

with (x,Y), (1/x,Y), (Y,x) et (Y, 1/x)

f1:=collect(subs(Nbis(x,y)=0,y=Y,eqNbis),[N,N10]); f2:=subs(x=1/x,f1);

LCwqMEkidEc2IiIiIiwmKiQpSSJZR0YlIiIjRiZGJkYmRiZGJiwmSSJ4R0YlRiZGJiEiIkYmLCZGLUYmRiZGJkYmRi1GLkYqRi4tSSJOR0YlNiQiIiFGKkYmRiYqLkYrRiZGJEYmLCYqJClGLUYrRiZGJkYmRiZGJkYtRi5GKkYuLUYxNiRGLUYzRiZGLioqRiRGJkYnRiZGKUYuLUYxNiRGKkYzRiZGLiooRiRGJkYqRi5JJE4xMEdGJUYmRiYqJkYrRiZGLUYmRiY=

LCwqMEkidEc2IiIiIiwmKiQpSSJZR0YlIiIjRiZGJkYmRiZGJiwmKiRJInhHRiUhIiJGJkYmRi9GJiwmRi1GJkYmRiZGJkYuRiZGKkYvLUkiTkdGJTYkIiIhRipGJkYmKi5GK0YmRiRGJiwmKiQpRi5GK0YvRiZGJkYmRiZGLkYmRipGLy1GMjYkRi1GNEYmRi8qKkYkRiZGJ0YmRilGLy1GMjYkRipGNEYmRi8qKEYkRiZGKkYvSSROMTBHRiVGJkYmKiZGK0YmRi5GL0Ym

f3:=collect(subs({x=y,y=x},Nbis(y,x)=0,y=Y,eqNbis),[N,N10]); f4:=subs(x=1/x,f3);

LCwqMEkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJiwmSSJZR0YlRiZGJiEiIkYmLCZGLUYmRiZGJkYmRi1GLkYqRi4tSSJOR0YlNiQiIiFGKkYmRiYqLkYrRiZGJEYmLCYqJClGLUYrRiZGJkYmRiZGJkYtRi5GKkYuLUYxNiRGLUYzRiZGLioqRiRGJkYnRiZGKUYuLUYxNiRGKkYzRiZGLiooRiRGJkYqRi5JJE4xMEdGJUYmRiYqJkYrRiZGLUYmRiY=

LCwqMEkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjISIiRiZGJkYmRiYsJkkiWUdGJUYmRiZGLEYmLCZGLkYmRiZGJkYmRi5GLEYqRiYtSSJOR0YlNiQiIiEqJEYqRixGJkYmKi5GK0YmRiRGJiwmKiQpRi5GK0YmRiZGJkYmRiZGLkYsRipGJi1GMTYkRi5GM0YmRiwqKkYkRiZGJ0YmRilGJi1GMTYkRjRGM0YmRiwqKEYkRiZGKkYmSSROMTBHRiVGJkYmKiZGK0YmRi5GJkYm

indets(f1*f2*f3*f4);

PCxJJE4xMEc2IkkiWUdGJEkidEdGJEkieEdGJC1JIk5HRiQ2JCIiIUYlLUYpNiRGK0YnLUYpNiRGKyokRichIiItRik2JEYlRistRik2JEYnRistRik2JEYwRis=

We eliminate the series involving Y

NYsol:=factor(solve({f1,f2},{N(0,Y),N(Y,0)}));

PCQvLUkiTkc2IjYkIiIhSSJZR0YmKiwsLiooLUYlNiRJInhHRiZGKCIiIkkidEdGJkYwKUYvIiIjRjBGMCooLUYlNiQqJEYvISIiRihGMEYxRjBGMkYwRjgqJkYyRjBGKUYwRjgqJkYxRjBGLUYwRjAqJkY1RjBGMUYwRjhGKUYwRjBGMUY4LCZGL0YwRjBGOEY4LCZGL0YwRjBGMEY4LCYqJClGKUYzRjBGMEYwRjBGOC8tRiU2JEYpRigqLCwwRixGOEY0RjgqKEYxRjBGL0YwSSROMTBHRiZGMEYwRjlGMEY6RjhGO0Y4RilGMEYwRilGMEYxRjhGL0Y4Rj5GOA==

f3bis:=collect(1/t/x/Y*numer(factor(subs(NYsol,f3))),[N,N10],factor);

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYsRi0tSSJOR0YoNiQiIiFGJ0YqRioqKEYkRipGJ0YtLUYwNiRGJ0YyRipGKioqRilGKkYkRipGJ0YtLUYwNiQqJEYnRi1GMkYqRipJJE4xMEdGKEYtKipGKUYqRixGKkkidEdGKEYtRidGLUYt

f4bis:=collect(1/t/x/Y*numer(factor(subs(NYsol,f4))),[N,N10],factor);

LCwqLiwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYnRi1GLEYtLUkiTkdGKDYkIiIhKiRGJ0YtRipGKiomLCYqJkYpRipGJkYqRipGKUYqRiotRjA2JEYnRjJGKkYqKiZGJEYqLUYwNiRGM0YyRipGKiomRidGKkkkTjEwR0YoRipGLSoqRilGKkYsRipJInRHRihGLUYmRipGLQ==

factor(subs(x=1/x,f3bis)/f4bis);

KiRJInhHNiIhIiI=

n:=10: normal(series(subs(N(0,x)=subs(x=0,y=x,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)), N10=coeff(subs(y=0,Nser(n)),x,1),Y=Y0ser(n+2),f3bis),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

Eliminate N(1/x,0)

f8:=collect((-f3bis+2/x*f4bis),[N,N10],factor);

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKkYsRi0tSSJOR0YoNiQiIiFGJ0YqRi0qMEYpRipGJkYtRiRGKkYrRipGLkYqRixGLS1GMDYkRjIqJEYnRi1GKkYqKioiIiRGKkYkRipGJ0YtLUYwNiRGJ0YyRipGKkkkTjEwR0YoRi0qLEYpRipGLEYqLCYqJkYpRipGJkYqRipGKkYtRipJInRHRihGLUYnRi1GLQ==

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

What we should remember from the above subsection

f8:=-(x^2+1)*(Y-1)*(Y+1)*N(0, x)/Y+(2*(x^2+1))*(Y-1)*(Y+1)*N(0, 1/x)/(x^2*Y)+(3*(x^2+1))*N(x, 0)/x-N10-2*Y*(2*x^2-1)/(t*x) ;

LCwqLCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiosJkkiWUdGKEYqRiohIiJGKiwmRixGKkYqRipGKi1JIk5HRig2JCIiIUYnRipGLEYtRi0qMEYpRipGJEYqRitGKkYuRiotRjA2JEYyKiRGJ0YtRipGJkYtRixGLUYqKioiIiRGKkYkRiotRjA2JEYnRjJGKkYnRi1GKkkkTjEwR0YoRi0qLEYpRipGLEYqLCYqJkYpRipGJkYqRipGKkYtRipGJ0YtSSJ0R0YoRi1GLQ==

factor(expand(rationalize(subs(Y=Y0sol,-(x^2+1)*(Y-1/Y)))));

KiZJInRHNiIhIiIpLCQqJiwoKigiIiMiIiJGI0YsKUkieEdGJEYrRixGLComRitGLEYjRixGLEYuRixGLCwoRipGLEYvRixGLkYlRixGJSNGLEYrRiw=

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

KiZJInhHNiIiIiJJInRHRiQhIiI=

Introducing De=Delta

f8form:=1/t*sqrt(De)*(x*N(0,x)-2/x*N(0,1/x))+3*(x+1/x)*N(x, 0)-2*Y*(2*x^2-1)/(x*t)-N10;

LCoqKEkidEc2IiEiIilJI0RlR0YlIyIiIiIiI0YqLCYqJkkieEdGJUYqLUkiTkdGJTYkIiIhRi5GKkYqKihGK0YqRi5GJi1GMDYkRjIqJEYuRiZGKkYmRipGKiooIiIkRiosJkYuRipGNkYqRiotRjA2JEYuRjJGKkYqKixGK0YqSSJZR0YlRiosJiomRitGKilGLkYrRipGKkYqRiZGKkYuRiZGJEYmRiZJJE4xMEdGJUYm

n:=10: normal(series(subs(De=Delta,N(0,x)=subs(x=0,y=x,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)), N10=coeff(subs(y=0,Nser(n)),x,1),Y=Y0ser(n+2),f8form),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

<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>

N(x,0) is a multiple of x, and N(x,0)/x is even in x and t

subs(y=0,Nser(4));

LChJInhHNiIiIiIqJiwmKiQpRiMiIiRGJUYlKiYiIiNGJUYjRiVGJUYlKUkidEdGJEYsRiVGJSomLCgqJkYsRiUpRiMiIiZGJUYlKiYiIilGJUYpRiVGJSomIiM2RiVGI0YlRiVGJSlGLiIiJUYlRiU=

The series xtN(0,x) is even in t and x

n:=9: normal(series(x*t*subs(x=0,y=x,Nser(n)),t,n+1));

Ky1JInRHNiIqJClJInhHRiQiIiMiIiJGKComRiZGKSwmKiYiIiRGKUYmRilGKSIiJkYpRikiIiUqJkYmRiksKComIiM1RikpRidGL0YpRikqJiIjT0YpRiZGKUYpIiNQRilGKSIiJyomRiZGKSwqKiYiI05GKSlGJ0Y4RilGKSomIiQmPkYpRjRGKUYpKiYiJDIlRilGJkYpRikiJEskRilGKSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGKUYz

Introduce the series R and S which are even in t and x (later we'll replace x by sqrt(x))

Rser:=proc(n) collect(t^2/x*subs(y=0,Nser(n-1)),t,factor): end:

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

Rser(4);Sser(4);

LCYqJiwmKiQpSSJ4RzYiIiIjIiIiRipGKUYqRiopSSJ0R0YoIiIlRipGKiokKUYsRilGKkYq

LCYqKClJInhHNiIiIiMiIiIsJiomIiIkRihGJEYoRigiIiZGKEYoKUkidEdGJiIiJUYoRigqJilGLkYnRihGJEYoRig=

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^2*x,N10=R0/t^2,Y=(1-sqrt(De))/2/t/(x+1/x),t^2*f8form),[De,R,S,R0],factor);

LCoqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjAqJiwmKiZGLEYqKUYpRixGKkYqRipGMEYqLCYqJEY0RipGKkYqRipGMEYqRiopSSNEZUdGJyNGKkYsRipGKiomLCYqJiIiJEYqRjRGKkYqRj1GKkYqLUkiUkdGJ0YoRipGKkkjUjBHRidGMEYxRjA=

n:=6: normal(series(subs(S(x)=Sser(n),S(1/x)=subs(x=1/x,Sser(n)),De=Delta,R(x)=Rser(n),R0=coeff(Rser(n),x,0),f13),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

op(1,f13);

KiYsKC1JIlNHNiI2I0kieEdGJiIiIiomIiIjRiktRiU2IyokRighIiJGKUYvKiYsJiomRitGKSlGKEYrRilGKUYpRi9GKSwmKiRGM0YpRilGKUYpRi9GKUYpKUkjRGVHRiYjRilGK0Yp

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

LCYqJiksKC1JIlNHNiI2I0kieEdGKCIiIiomIiIjRistRic2IyokRiohIiJGK0YxKiYsJiomRi1GKylGKkYtRitGK0YrRjFGKywmKiRGNUYrRitGK0YrRjFGK0YtRitJI0RlR0YoRitGKyokKSwoKiYsJiomIiIkRitGNUYrRitGP0YrRistSSJSR0YoRilGK0YrSSNSMEdGKEYxRjJGMUYtRitGMQ==

u1:=collect(u0*(x^2+1),[S,R0,R],distributed,factor);

LDgqKEkjRGVHNiIiIiIsJiokKUkieEdGJSIiI0YmRiZGJkYmRiYpLUkiU0dGJTYjRipGK0YmRiYqLCIiJUYmRiRGJkYnRiZGLUYmLUYuNiMqJEYqISIiRiZGNSoqRitGJiwmKiZGK0YmRilGJkYmRiZGNUYmRiRGJkYtRiZGJioqRjFGJkYkRiZGJ0YmKUYyRitGJkYmKipGMUYmRjdGJkYkRiZGMkYmRjUqJiwmRihGNUYmRjVGJilJI1IwR0YlRitGJkYmKioiIidGJilGJ0YrRiZGP0YmLUkiUkdGJUYvRiZGJiomLCYqJkYxRiZGKUYmRjVGK0YmRiZGP0YmRiYqKCIiKkYmKUYnIiIkRiYpRkNGK0YmRjUqKkZBRiZGJ0YmRjdGJkZDRiZGJiooKUY3RitGJiwmRiRGJkYmRjVGJkYnRjVGJg==

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

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

use parity: S(x,t)=Ss(x^2,t), same for R. We keep t instead of sqrt(t), even though there is parity in t as well. The series Rs and Ss are denoted R and S in the paper.

u1expls:=collect(expand(subs( R0=Rs0,x=sqrt(x),S(sqrt(x))=Ss(x),S(1/sqrt(x))=Ss(1/x), R(sqrt(x))=Rs(x),u1expl)),[Ss,Rs,Rs0],distributed,factor);

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

Deltas:=collect(subs(x=sqrt(x),expand(Delta)),t,factor);

LCYiIiJGIyoqIiIlRiMpLCZJInhHNiJGI0YjRiMiIiNGI0YoISIiKUkidEdGKUYqRiNGKw==

Rsser:=proc(n) subs(x=sqrt(x),Rser(2*n)): end:

Rsser(3);

LCgqJiwoKiYiIiMiIiIpSSJ4RzYiRiZGJ0YnKiYiIilGJ0YpRidGJyIjNkYnRicpSSJ0R0YqIiInRidGJyomLCZGKUYnRiZGJ0YnKUYvIiIlRidGJyokKUYvRiZGJ0Yn

Ssser:=proc(n) subs(x=sqrt(x),Sser(2*n)): end:

Ssser(3);

LCgqKEkieEc2IiIiIiwoKiYiIzVGJilGJCIiI0YmRiYqJiIjT0YmRiRGJkYmIiNQRiZGJilJInRHRiUiIidGJkYmKihGJEYmLCYqJiIiJEYmRiRGJkYmIiImRiZGJilGMCIiJUYmRiYqJilGMEYrRiZGJEYmRiY=

series(u1expls,x=-1,1);

KycsJkkieEc2IiIiIkYmRiYsJiomIiInRiZJJFJzMEdGJUYmRiYqJkYpRiYtSSNTc0dGJTYjISIiRiZGJkYvLUkiT0clKnByb3RlY3RlZEc2I0YmIiIh

An identity that will be useful later

Rs0sol:=-Ss(-1):

Extract the negative part in x (which must be zero). Careful with the denominators (x+1)

u1expls;

We must be a bit careful with the terms in 1/(x+1)

u1n:=op(3,u1expls)+op(9,u1expls)-12*Ss(-1)/(x+1);

LCgqKiIiJSIiIiwqKihGJEYlKUkidEc2IiIiI0YlKUkieEdGKkYrRiVGJSooIiIpRiVGKEYlRi1GJUYlKiZGJEYlRihGJUYlRi0hIiJGJUYtRjEpLUkjU3NHRio2IyokRi1GMUYrRiVGMSouRiRGJSwmKiZGK0YlRi1GJUYlRiVGMUYlRiZGJUYtRjEsJkYtRiVGJUYlRjFGM0YlRiUqKCIjN0YlLUY0NiNGMUYlRjpGMUYx

series(u1n,x=-1,2);

KycsJkkieEc2IiIiIkYmRiYsKComIiIlRiYpLUkjU3NHRiU2IyEiIiIiI0YmRiYqJiIjN0YmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsRi1GJkYuKiYiIilGJkYrRiZGLiIiIS1JIk9HRjY2I0YmRiY=

u1p:=subs(Ss(1/x)=0,u1expls-u1n);

LDQqKiIiJSIiIiksJiomIiIjRiVJInhHNiJGJUYlRiUhIiJGKUYlKUkidEdGK0YpRiVGKkYsRiwqKCwqKihGJEYlRi1GJSlGKkYpRiVGJSooIiIpRiVGLUYlRipGJUYlKiZGJEYlRi1GJUYlRipGLEYlRipGLCktSSNTc0dGKzYjRipGKUYlRiwqKCIiKkYlKSwmRipGJUYlRiVGKUYlKS1JI1JzR0YrRjlGKUYlRiwqJClJJFJzMEdGK0YpRiVGLCoqRilGJUYnRiVGPUYsRkNGJUYsKiYsJiIiJ0YsKiYiIzdGJUYqRiVGJUYlRj9GJUYlKi5GKUYlRidGJUYwRiVGKkYsRj1GLEY3RiVGLCooLCYqJkZHRiVGKkYlRiVGR0YlRiVGP0YlRkNGJUYlKihGSUYlLUY4NiNGLEYlRj1GLEYl

Terms with no series in 1/x

subs(Ss(0)=0,series(u1p,x,1));

KydJInhHNiIsJComIiIlIiIiKUkidEdGJCIiI0YoISIiRiwtSSJPRyUqcHJvdGVjdGVkRzYjRigiIiE=

The parts with S(1/x)

factor(subs(Ss(0)=0,series(subs(x=1/x,u1n),x,3)));

KylJInhHNiIsJCooIiNLIiIiKUkidEdGJCIiI0YoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kjU3NHRiQ2IyIiIUYoRihGNSwsKigiIztGKEYpRigpRixGK0YoISIiKihGOEYoRilGKC0tLUkjQEBHRi82JEYuRitGMkY0RihGKCooRjhGKEYpRihGLEYoRigqJiIiKUYoRixGKEY6KiYiIzdGKC1GMzYjRjpGKEY6RigtSSJPR0YwNiNGKEYr

The negative part:

u2:=-4*Sneg(1/x)-4*t^2/x+u1n-coeff(%,x,0);

LC4qJiIiJSIiIi1JJVNuZWdHNiI2IyokSSJ4R0YoISIiRiVGLCooRiRGJUYrRiwpSSJ0R0YoIiIjRiVGLCoqRiRGJSwqKihGJEYlRi5GJSlGK0YwRiVGJSooIiIpRiVGLkYlRitGJUYlKiZGJEYlRi5GJUYlRitGLEYlRitGLCktSSNTc0dGKEYpRjBGJUYsKi5GJEYlLCYqJkYwRiVGK0YlRiVGJUYsRiVGMkYlRitGLCwmRitGJUYlRiVGLEY5RiVGJSooIiM3RiUtRjo2I0YsRiVGPkYsRiwqKCIjS0YlRi5GJS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiNGOjYjIiIhRiVGLA==

series(u2,x=-1,2);

KycsJkkieEc2IiIiIkYmRiYsLiomIiIlRiYtSSVTbmVnR0YlNiMhIiJGJkYtKiZGKUYmKUkidEdGJSIiI0YmRiYqJkYpRiYpLUkjU3NHRiVGLEYxRiZGJiomIiM3RiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRjVGLEYmRi0qJiIiKUYmRjRGJkYtKigiI0tGJkYvRiYtRjk2IyIiIUYmRi1GRS1JIk9HRjw2I0YmRiY=

let us check: this must be non-negative

n:=10: factor(series(subs((D(Ss))(0)=coeff(Ssser(n),x,1),Sneg(1/x)=Deltas*Ss(x)*Ss(1/x),Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)) , Ss(-1)=subs(x=-1,Ssser(n)), u2),t,n));

KytJInRHNiIhIiUiIiUsJiomRiYiIiJJInhHRiRGKUYpIiIpISIiIiInLCgqJkYrRikpRioiIiNGKUYpKiYiI19GKUYqRilGKSIjO0YsRistSSJPRyUqcHJvdGVjdGVkRzYjRikiIzU=

and this must be negative

n:=16: expand(factor(series(subs((D(Ss))(0)=coeff(Ssser(n),x,1),Sneg(1/x)=0,Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)) , Ss(-1)=subs(x=-1,Ssser(n)), u2),t,n)));

Ky9JInRHNiIsJComIiIlIiIiSSJ4R0YkISIiRioiIicsJiomIiNfRihGKUYqRioqJiIiKUYoKUYpIiIjRipGKkYwLCgqJiIkMydGKEYpRipGKiomIiRjIkYoRjFGKkYqKiYiIz9GKClGKSIiJEYqRioiIzUsKiomIiVfckYoRilGKkYqKiYiJWNCRihGMUYqRioqJiIkSyZGKEY6RipGKiomIiNjRigpRilGJ0YqRioiIzcsLComIiZfaClGKEYpRipGKiomIiY/SSRGKEYxRipGKiomIiUhKSoqRihGOkYqRioqJiIlLz5GKEZGRipGKiomIiRvIkYoKUYpIiImRipGKiIjOS1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjOw==

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

LC4tSSVTbmVnRzYiNiMqJEkieEdGJSEiIkYpKiZGKEYpKUkidEdGJSIiIyIiIkYpKigsKiooIiIlRi5GK0YuKUYoRi1GLkYuKigiIilGLkYrRi5GKEYuRi4qJkYyRi5GK0YuRi5GKEYpRi5GKEYpKS1JI1NzR0YlRiZGLUYuRikqLCwmKiZGLUYuRihGLkYuRi5GKUYuRjBGLkYoRiksJkYoRi5GLkYuRilGOEYuRi4qKCIiJEYuLUY5NiNGKUYuRj1GKUYpKihGNUYuRitGLi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGOTYjIiIhRi5GKQ==

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

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

LDQqKEkkRGVzRzYiIiIiLUkjU3NHRiU2I0kieEdGJUYmLUYoNiMqJEYqISIiRiZGJiomIiIkRiYtRig2I0YuRiZGJiooLCoqKCIiJUYmKUkidEdGJSIiI0YmKUYqRjlGJkYmKigiIilGJkY3RiZGKkYmRiYqJkY2RiZGN0YmRiZGKkYuRiZGKkYuKUYnRjlGJkYmKiwsJkYqRiZGOUYuRiZGNEYmRipGLiwmRipGJkYmRiZGLkYnRiZGJiooRjRGJkYqRi4pRitGOUYmRiYqLCwmKiZGOUYmRipGJkYmRiZGLkYmRjRGJkYqRi5GQUYuRitGJkYuKigiIztGJkY3RiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRig2IyIiIUYmRiZJI1AwR0YlRi4qKCwmKiRGOkYmRiZGJkYmRiZGN0YmRipGLkYm

Let us check

n:=15: factor(series(subs((D(Ss))(0)=coeff(Ssser(n),x,1),Des=Deltas,Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)), P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0), Ss(-1)=subs(x=-1,Ssser(n)), u5),t,n));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

series(subs(Des=Deltas,u5),x=-1,2);;

KycsJkkieEc2IiIiIkYmRiYsLiokKS1JI1NzR0YlNiMhIiIiIiNGJkYtKiYiIiVGJkYqRiZGJiomIiInRiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRitGLEYmRiYqKCIjO0YmKUkidEdGJUYuRiYtRjQ2IyIiIUYmRiZJI1AwR0YlRi0qJkYuRiZGPEYmRi1GQC1JIk9HRjc2I0YmRiY=

What to remember from the previous sub-section

Rs0sol:=-Ss(-1);

LCQtSSNTc0c2IjYjISIiRic=

u5 := Des*Ss(x)*Ss(1/x)+3*Ss(-1)+(4*t^2*x^2+8*t^2*x+4*t^2-x)*Ss(x)^2/x+(x-2)*(4*t^2*x^2+8*t^2*x+4*t^2-x)*Ss(x)/(x*(x+1))+(4*t^2*x^2+8*t^2*x+4*t^2-x)*Ss(1/x)^2/x-(2*x-1)*(4*t^2*x^2+8*t^2*x+4*t^2-x)*Ss(1/x)/(x*(x+1))+16*t^2*(D(Ss))(0)-P0+(x^2+1)*t^2/x;

u5form := (Ss(x)*Ss(1/x)-Ss(x)^2-(x-2)*Ss(x)/(x+1)-Ss(1/x)^2+(2*x-1)*Ss(1/x)/(x+1))*Des+3*Ss(-1)+16*t^2*(D(Ss))(0)-P0+(x^2+1)*t^2/x;

LCwqJiwsKiYtSSNTc0c2IjYjSSJ4R0YoIiIiLUYnNiMqJEYqISIiRitGKyokKUYmIiIjRitGLyooLCZGKkYrRjJGL0YrRiZGKywmRipGK0YrRitGL0YvKiQpRixGMkYrRi8qKCwmKiZGMkYrRipGK0YrRitGL0YrRixGK0Y1Ri9GK0YrSSREZXNHRihGK0YrKiYiIiRGKy1GJzYjRi9GK0YrKigiIztGKylJInRHRihGMkYrLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0YnNiMiIiFGK0YrSSNQMEdGKEYvKigsJiokKUYqRjJGK0YrRitGK0YrRkJGK0YqRi9GKw==

Deltas:=collect(subs(x=sqrt(x),expand(Delta)),t,factor);

LCYiIiJGIyoqIiIlRiMpLCZJInhHNiJGI0YjRiMiIiNGI0YoISIiKUkidEdGKUYqRiNGKw==

factor(subs(Des=Deltas,u5form/u5));

IiIi

Rser:=proc(n) collect(t^2/x*subs(y=0,Nser(n-1)),t,factor): end:

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

Rsser:=proc(n) subs(x=sqrt(x),Rser(n)): end:

Ssser:=proc(n) subs(x=sqrt(x),Sser(n)): end:

Ssser(5);

LCgqKEkieEc2IiIiIiwoKiYiIzVGJilGJCIiI0YmRiYqJiIjT0YmRiRGJkYmIiNQRiZGJilJInRHRiUiIidGJkYmKihGJEYmLCYqJiIiJEYmRiRGJkYmIiImRiZGJilGMCIiJUYmRiYqJilGMEYrRiZGJEYmRiY=

n:=10: normal(series(subs(Ss(x)=Ssser(n),Ss(1/x)=subs(x=1/x,Ssser(n)),Des=Deltas,Ss(-1)=subs(x=-1,Ssser(n)),D(Ss)(0)=coeff(Ssser(n),x,1),P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0),u5form),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

<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*(-1+Ss(x)+Ss(1/x))),[Des,Ss,P0],factor);

LDAqJiwuKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjIqJCktRig2IyokRitGMkYsRi1GLSoqRixGLSlGN0YwRi1GMUYyRitGLUYyKigsJiomRjBGLUYrRi1GLUYtRjJGLUY3Ri1GMUYyRi1GLUkkRGVzR0YpRi1GLSomLCgqJkYsRi1GJ0YtRjIqJkYsRi1GN0YtRjJGLEYtRi0tRig2I0YyRi1GLSomLCgqKCIjO0YtKUkidEdGKUYwRi0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjJJI1AwR0YpRi0qKCwmKiQpRitGMEYtRi1GLUYtRi1GSkYtRitGMkYyRi1GJ0YtRi0qJkZHRi1GN0YtRi1GSEYtRlVGMkZWRi0=

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

IiIi

factor(series(subs(Des=Deltas,u11),x=-1,2));

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Extract the non-negative part, being careful with the divisions by 1/(1+x)

u11p:=subs(Ss(1/x)=0,u11)+3*Ss(-1)*(Ss(-1)-1)/(x+1);

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factor(series(subs(Des=Deltas,u11p),x=-1,1));

KycsJkkieEc2IiIiIkYmRiYsOCoqIiM7RiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSNTc0dGJTYjIiIhRiYtRjE2IyEiIkYmKUkidEdGJSIiI0YmRjYqKEYpRiZGN0YmRipGJkYmKiQpRjQiIiRGJkYmKihGOUYmRjRGJkY3RiZGJiomRj1GJilGNEY5RiZGNiooIiInRiZGNEYmLUYrRjVGJkY2KiZGNEYmSSNQMEdGJUYmRiYqJkY5RiZGN0YmRjYqJkY5RiZGNEYmRiYqJkY9RiZGQ0YmRiZGRUY2RjMtSSJPR0YuNiNGJkYm

u11n:=collect(u11-u11p,[Des,Ss],factor);

LCoqJiwoKiQpLUkjU3NHNiI2IyokSSJ4R0YpISIiIiIkIiIiRi8qKkYuRi8pRiciIiNGLywmRixGL0YvRi9GLUYsRi9GLSooLCYqJkYyRi9GLEYvRi9GL0YtRi9GJ0YvRjNGLUYvRi9JJERlc0dGKUYvRi8qKEYuRi9GM0YtKS1GKDYjRi1GMkYvRi0qJiwmKiZGLkYvRidGL0YtKiZGLkYvRjNGLUYvRi9GOkYvRi8qJiwmKigiIztGLylJInRHRilGMkYvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2I0YoNiMiIiFGL0YtKiYsKComRkRGLylGLEYyRi9GLSomSSNQMEdGKUYvRixGL0YvKiRGREYvRi1GL0YsRi1GL0YvRidGL0Yv

A part to be substracted

normal(subs(Ss(0)=0,series(subs(Des=Deltas,u11p),x,1)));

KydJInhHNiIqJClJInRHRiQiIiMiIiIhIiItSSJPRyUqcHJvdGVjdGVkRzYjRikiIiE=

factor(subs(Ss(0)=0,series(subs(x=1/x,Des=Deltas,u11n),x,3)));

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eqSp:=collect(u11p-convert(%%,polynom)+convert(%,polynom),[Des,Ss(x),x],factor);

LCgqJiwoKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjJGLUkkRGVzR0YpRi1GLSomLCwqJilJInRHRilGMEYtRitGLUYyKigiIztGLUY5Ri0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjIqJkYsRi0tRig2I0YyRi1GMkkjUDBHRilGLSomRitGMkY5Ri1GMkYtRidGLUYtKiYsMioqIiIoRi1GOUYtRj1GLUYrRi1GLSomRjlGLSlGK0YwRi1GLSooRk5GLUY5Ri1GPUYtRi1GOEYtKiZGLEYtKUZHRjBGLUYtKihGLEYtRkdGLUYrRi1GLSomRklGLUYrRi1GMkZJRjJGLUYxRjJGLQ==

Let us check this equation

n:=10: normal(series(subs(Des=Deltas,D(Ss)(0)=coeff(Ssser(n),x,1),Ss(-1)=subs(x=-1,Ssser(n)),Ss(0)=subs(x=0,Ssser(n)),Ss(x)=Ssser(n),P0=coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0),eqSp),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

How many unknown series do we really have? Let us rewrite the constant term

map(factor,convert(subs(Ss(x)=0,eqSp),parfrac,x));

LCwqJilJInRHNiIiIiMiIiJJInhHRiZGKEYoKigiIihGKEYkRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSNTc0dGJjYjIiIhRihGKComIiIkRigtRjM2IyEiIkYoRihJI1AwR0YmRjoqKkY3RihGOEYoLCZGOEYoRihGOkYoLCZGKUYoRihGKEY6Rig=

eqSp1:=(Ss(x)^3-3*Ss(x)^2/(x+1)-(x-2)*Ss(x)/(x+1))*Des+(-t^2*x-16*t^2*(D(Ss))(0)-3*Ss(-1)+P0-t^2/x)*Ss(x)+%;

LDAqJiwoKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjJGLUkkRGVzR0YpRi1GLSomLCwqJilJInRHRilGMEYtRitGLUYyKigiIztGLUY5Ri0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjIqJkYsRi0tRig2I0YyRi1GMkkjUDBHRilGLSomRitGMkY5Ri1GMkYtRidGLUYtRjhGLSooIiIoRi1GOUYtRj1GLUYtRkZGLUZJRjIqKkYsRi1GR0YtLCZGR0YtRi1GMkYtRjFGMkYt

factor(eqSp-eqSp1);

IiIh

Introduce F0 instead of P0

eqSp2:=subs(P0=F0+3*Ss(-1),eqSp1);

LC4qJiwoKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjJGLUkkRGVzR0YpRi1GLSomLCoqJilJInRHRilGMEYtRitGLUYyKigiIztGLUY5Ri0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjJJI0YwR0YpRi0qJkYrRjJGOUYtRjJGLUYnRi1GLUY4Ri0qKCIiKEYtRjlGLUY9Ri1GLUZGRjIqKkYsRi0tRig2I0YyRi0sJkZLRi1GLUYyRi1GMUYyRi0=

F0ser:=proc(n) series(coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0)-3*subs(x=-1,Ssser(n)),t,n+1): end:

F0ser(8);

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Expansion at x=0

ct:=coeff(factor(subs(Ss(0)=0,series(subs(Des=Deltas,eqSp2),x,2))),x,0);

LCoqKCIiIyIiIilJInRHNiJGJEYlLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0kjU3NHRig2IyIiIUYlISIiKiYiIiRGJSktRjA2I0YzRiRGJUYlSSNGMEdGKEYzKiZGNUYlRjdGJUYz

This allows to express the last term in eqSp2 in terms of S1 and F0

eqSp3:=(-(x-2)*Ss(x)/(x+1)+Ss(x)^3-3*Ss(x)^2/(x+1))*Des+(-t^2*x-16*t^2*(D(Ss))(0)+F0-t^2/x)*Ss(x)+7*t^2*(D(Ss))(0)+t^2*x-F0 +(F0+2*t^2*D(Ss)(0))/(x+1);

LC4qJiwoKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjJGLUkkRGVzR0YpRi1GLSomLCoqJilJInRHRilGMEYtRitGLUYyKigiIztGLUY5Ri0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjJJI0YwR0YpRi0qJkYrRjJGOUYtRjJGLUYnRi1GLSooIiIoRi1GOUYtRj1GLUYtRjhGLUZGRjIqJiwmRkZGLSooRjBGLUY5Ri1GPUYtRi1GLUYxRjJGLQ==

n:=7: normal(series(subs(Des=Deltas,Ss(x)=Ssser(n),D(Ss)(0)=coeff(Ssser(n),x,1), F0=F0ser(n),eqSp3),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

What needs to be remembered from this subsection

F0ser:=proc(n) series(coeff(Deltas*Ssser(n)*subs(x=1/x,Ssser(n)),x,0)-3*subs(x=-1,Ssser(n)),t,n+1): end:

The equation satisfied by Ss

eqSp3:=(Ss(x)^3-3*Ss(x)^2/(x+1)-(x-2)*Ss(x)/(x+1))*Des+(-t^2*x-16*t^2*(D(Ss))(0)+F0-t^2/x)*Ss(x)+7*t^2*(D(Ss))(0)+t^2*x-F0+(F0+2*t^2*(D(Ss))(0))/(x+1);

LC4qJiwoKiQpLUkjU3NHNiI2I0kieEdGKSIiJCIiIkYtKihGLEYtKUYnIiIjRi0sJkYrRi1GLUYtISIiRjIqKCwmRitGLUYwRjJGLUYnRi1GMUYyRjJGLUkkRGVzR0YpRi1GLSomLCoqJilJInRHRilGMEYtRitGLUYyKigiIztGLUY5Ri0tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjRig2IyIiIUYtRjJJI0YwR0YpRi0qJkYrRjJGOUYtRjJGLUYnRi1GLSooIiIoRi1GOUYtRj1GLUYtRjhGLUZGRjIqJiwmRkZGLSooRjBGLUY5Ri1GPUYtRi1GLUYxRjJGLQ==

n:=20: normal(series(subs(Des=Deltas,Ss(x)=Ssser(n),D(Ss)(0)=coeff(Ssser(n),x,1), F0=F0ser(n),eqSp3),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNB

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

eqS:=map(factor,subs(Des=Deltas,Ss(x)=Ssx,D(Ss)(0)=S1,eqSp3));

LC4qLkkkU3N4RzYiIiIiLCZGJEYmRiYhIiJGJiwqKiZGJEYmSSJ4R0YlRiZGJkYkRiZGK0YmIiIjRihGJiwqKigiIiVGJilJInRHRiVGLEYmKUYrRixGJkYmKigiIilGJkYwRiZGK0YmRiYqJkYvRiZGMEYmRiZGK0YoRiZGK0YoLCZGK0YmRiZGJkYoRigqKCwqKioiIztGJkkjUzFHRiVGJkYwRiZGK0YmRigqJkYwRiZGMkYmRigqJkkjRjBHRiVGJkYrRiZGJiokRjBGJkYoRiZGJEYmRitGKEYmKigiIihGJkYwRiZGO0YmRiYqJkYwRiZGK0YmRiZGPkYoKiYsJiooRixGJkY7RiZGMEYmRiZGPkYmRiZGNkYoRiY=

Differentiating with respect to x_0

eqSS:=diff(eqS,Ssx);

LCoqLCwmSSRTc3hHNiIiIiJGJyEiIkYnLCoqJkYlRidJInhHRiZGJ0YnRiVGJ0YrRiciIiNGKEYnLCoqKCIiJUYnKUkidEdGJkYsRicpRitGLEYnRicqKCIiKUYnRjBGJ0YrRidGJyomRi9GJ0YwRidGJ0YrRihGJ0YrRigsJkYrRidGJ0YnRihGKCosRiVGJ0YpRidGLUYnRitGKEY2RihGKCoqRiVGJ0YkRidGLUYnRitGKEYoKiYsKioqIiM7RidJI1MxR0YmRidGMEYnRitGJ0YoKiZGMEYnRjJGJ0YoKiZJI0YwR0YmRidGK0YnRicqJEYwRidGKEYnRitGKEYn

How many series Xi cancel eqSS? Two

collect(factor(subs(Ssx=t*x*J,S1=t*J1,numer(eqSS))),t);

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F0ser(3);

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subs(t=0,Ssx=0,F0=0,eqSS*x*(x+1));

LCQqJiwmSSJ4RzYiIiIiIiIjISIiRidGJUYnRik=

n:=20:expr:=normal(series(subs(Ssx=Ssser(n),F0=F0ser(n),S1=coeff(Ssser(n),x,1), eqSS*x*(x+1)),t,n+1)):

with(gfun):Xisol:=algeqtoseries(convert(expr,polynom),t,x,12,true);

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ifactor([117,3147,5067]);

NyUqJiktSSFHNiI2IyIiJCIiIyIiIi1GJjYjIiM4RisqJkYlRistRiY2IyIlXDVGKyomRiRGKy1GJjYjIiRqJkYr

What are these roots...we'll see later that they have degree 16 over Q(t)

Differentiating w.r.t. x

eqSx:=collect(diff(eqS,x),[Ssx,F0,S1],factor);

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Let us check that eqSx also vanishes

normal(series(subs(Ssx=Ssser(n), F0=F0ser(n),S1=coeff(Ssser(n),x,1), x=op(1,Xisol),eqSx),t,10));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

normal(series(subs(Ssx=Ssser(n), F0=F0ser(n),S1=coeff(Ssser(n),x,1), x=op(2,Xisol),eqSx),t,10));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIp

<Text-field style="Heading 3" layout="Heading 3">Equations for S1 and P0 using double roots of the discriminant </Text-field>

factor(discrim(eqS,Ssx));

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

The big factor

disc:=collect(factor(numer(factor(1/Deltas*discrim(eqS,Ssx)))),x,factor);

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

let us write disc/x^4 as a polynomial (of degree 4) in s=x+1/x

reste:=disc/x^4: discs:=0:

for i from 4 to 0 by -1 do discs:=discs+factor(coeff(reste,x,i))*s^i:reste:=expand(reste-(x+1/x)^i*coeff(reste,x,i)):od:

discs;

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

This polynomial in s has two double roots, and hence is the square of a quadratic polynomial. We read off the constant term above

discf:=-81*t^4*(-40*S1*t^2-16*t^2+4*F0-1)*(s^2+a*s+b)^2;

LCQqKiIjIikiIiIpSSJ0RzYiIiIlRiUsKiooIiNTRiVJI1MxR0YoRiUpRiciIiNGJSEiIiomIiM7RiVGLkYlRjAqJkYpRiVJI0YwR0YoRiVGJUYlRjBGJSksKComSSJhR0YoRiVJInNHRihGJUYlKiQpRjlGL0YlRiVJImJHRihGJUYvRiVGMA==

We obtain a system of 4 equations (with 4 unknowns S1, F0, a and b) by saying that this is discs

eqens:={coeffs(collect(discs-discf,s,factor),s)};nops(%);

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

IiIl

The first equation allows us to express a in terms of S1 and F0

asol:=factor(isolate(eqens[1],a));

L0kiYUc2IiwkKiojIiIiIiIqRigsNiooIiQ9I0YoKUkjUzFHRiQiIiNGKClJInRHRiQiIiVGKCEiIiooIiQ7KEYoRi5GKEYwRihGKCoqIiM/RihJI0YwR0YkRihGLkYoKUYxRi9GKEYzKiYiJCdHRihGMEYoRigqKCIjIypGKEY4RihGOUYoRjMqKCIjJilGKEYuRihGOUYoRjMqJkYyRigpRjhGL0YoRigqJiIjUEYoRjlGKEYzKiYiIihGKEY4RihGKEYvRjNGKCwqKigiI1NGKEYuRihGOUYoRjMqJiIjO0YoRjlGKEYzKiZGMkYoRjhGKEYoRihGM0YzRjlGM0Yz

The second equation gives similarly b^2

b2sol:=factor(isolate(eqens[2],b^2));

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

the third equation gives an alternative expression of b

numer(factor(subs(asol,eqens[3])));

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

bsol:=factor(isolate(%,b));

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

The fourth equation still gives a new expression of b

numer(factor(subs(asol,eqens[4])));

LCQqKCIiJyIiIilJInRHNiIiIiNGJSxccCooIicxODVGJSlJI1MxR0YoIiIlRiUpRidGJEYlISIiKigiJVNDRiUpRi4iIiRGJUYwRiVGMSoqIiYrSyVGJSlGLkYpRiVJImJHRihGJUYwRiVGJSoqIiZjRyRGJUkjRjBHRihGJUY0RiUpRidGL0YlRiUqKCImN3kqRiVGOEYlRjBGJUYlKioiJmdYJEYlRi5GJUY5RiVGMEYlRiUqKiIkYyVGJUY8RiVGOEYlRj1GJUYlKiwiJVMnKUYlRjxGJUYuRiVGOUYlRj1GJUYxKigiJTkjKUYlRjRGJUY9RiVGMSooIiYpR25GJUYuRiVGMEYlRiUqKCIlN3BGJUY5RiVGMEYlRiUqKiIlX05GJSlGPEYpRiVGOEYlRiZGJUYxKioiJmMlPUYlRjxGJUYuRiVGPUYlRjEqKiIlY01GJUY8RiVGOUYlRj1GJUYxKigiJCE+RiVGOEYlRj1GJUYlKioiJWdARiVGLkYlRjlGJUY9RiVGJSomIiY1UiJGJUYwRiVGJSoqIiQpR0YlRk5GJUYuRiVGJkYlRiUqKiIkSyVGJUZORiVGOUYlRiZGJUYlKioiJXc8RiVGPEYlRjhGJUYmRiVGJSooIiUjeidGJUY8RiVGPUYlRjEqKCIlcVNGJUYuRiVGPUYlRiUqKCIkaylGJUY5RiVGPUYlRiUqKCIkRyJGJSlGPEY1RiVGLkYlRiUqKCIkZypGJUZORiVGJkYlRiUqKiIkZyZGJUY8RiVGLkYlRiZGJUYxKioiJDsjRiVGPEYlRjlGJUYmRiVGMSooIiRBI0YlRjhGJUYmRiVGMSomIiU5PEYlRj1GJUYlKiYiI0tGJUZhb0YlRjEqKCIjJypGJUZORiVGLkYlRjEqKCIkMydGJUY8RiVGJkYlRjEqKCIkKWZGJUYuRiVGJkYlRjEqKCIjRkYlRjlGJUYmRiVGJSomIiNrRiVGTkYlRiUqKCIjQ0YlRjxGJUYuRiVGJSomIiQrIkYlRiZGJUYxKiYiI1lGJUY8RiVGJSomRilGJUYuRiVGMSIjOkYxRiVGJQ==

bsolbis:=factor(isolate(%,b));

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

Comparing our two expressions of b gives an equation between S1 and F0

e1:=numer(factor(op(2,bsol)-op(2,bsolbis)));

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

Comparing the expression of b and b^2 gives a second equation

e2:=numer(factor(op(2,bsol)^2-op(2,b2sol)));

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

Now we eliminate F0 to obtain an equation for S1

res:=factor(resultant(e1,e2,F0));

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

n:=10: for i from 1 to nops(res) do series(subs(S1=coeff(Ssser(n),x,1),op(i,res)),t,n+1); od;

KyVJInRHNiIhOVdkSC55dU9XWVYjSCIiIiE=

KyVJInRHNiIiIiIiIiU=

Ky9JInRHNiIhIiIiIiEiIiIiIiMiIiYiIiUiI1AiIiciJEskIiIpIiVWTCIjNQ==

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KzFJInRHNiIiJEgoIiIhISZHTCMiIiMiJ0kkPSQiIiUhKE94bSIiIichKVQiZkwiIiIpIiopM1AnPiQiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KzFJInRHNiIiI2siIiEiJFciIiIjISQhPSIiJSElLDwiIichJiEqMyIiIikhJiw8KiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

KzFJInRHNiIhI0YiIiEiJSEzIiIiIyEmOEEiIiIlIiYnKWUiIiInIidmWTciIikiJ1dkeiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

KzFJInRHNiIiIiIiIiEhIyEqIiIjIiVITSIiJSEmKVFyIiInIid5KHApIiIpISgxXzMnIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

algS1:=collect(op(4,res),S1,factor);

LCwqKCIkSCgiIiIpSSNTMUc2IiIiJUYlKUkidEdGKCIiJ0YlRiUqKiIkViNGJSlGK0YpRiUsJiomRilGJSlGKyIiI0YlRiVGJUYlRiUpRiciIiRGJUYlKioiI0ZGJUYyRiUsKComIiM5RiVGL0YlRiUqJiIjPkYlRjJGJUYlRiUhIiJGJSlGJ0YzRiVGPSoqLCYqJiIjP0YlRjJGJUYlRiVGPUYlLCgqJiIiKEYlRjJGJUYlKiZGLEYlRitGJUY9RiVGJUYlLChGREYlRkZGJUYlRiVGJUYnRiVGPSomRjJGJSwoKiYiJFYkRiVGL0YlRiUqJiIjUEYlRjJGJUY9RiVGJUYlRj0=

Now we compute an equation for F0

res:=factor(resultant(e1,algS1,S1)): nops(res);

IiIl

n:=10: for i from 1 to nops(res) do series(subs(F0=F0ser(n),op(i,res)),t,n+1); od;

KyVJInRHNiIiIzsiIiE=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNJ

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KzFJInRHNiIiNCtnRFsjNDEqKmY8IiIhITZzIWZNd2wxYiIpSHAiIiMiOU9ad3FJZ1dJbCNRMyIiIiUhOnosJnliTDAvJz1AJVInKiIiJyI8I1J2MlFwTnZgMmxMInAmIiIpIT5nJnkmNCgpKWUkeURjdXZLVCMiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

op(2,res);

KiQpSSJ0RzYiIiNJIiIi

alfF0:=collect(op(3,res),F0,factor);

LCwqJiIkSCgiIiIpSSNGMEc2IiIiJUYlRiUqJiwmKiYiJVc+RiUpSSJ0R0YoIiIjRiUhIiIiJSw8RiVGJSlGJyIiJEYlRiUqJiwoKiYiJiNSU0YlKUYvRilGJUYxKiYiJlkvJUYlRi5GJUYxIiVCOEYlRiUpRidGMEYlRiUqKiwmKiYiJFsjRiVGLkYlRiUiIihGMUYlLCgqJiIjQUYlRi5GJUYlKiYiI1VGJUYvRiVGMUZCRiVGJSwoRkRGJUZGRiVGQkYlRiVGJ0YlRjEqJkYuRiUsKiomIichWzEiRiUpRi8iIidGJUYlKiYiJy9sb0YlRjlGJUYlKiYiJkNGJkYlRi5GJUYxIiVINUYlRiVGMQ==

Another approach: use generalized resultants to say that discs and its derivative w.r.t. s have two roots in common. First, the discriminant of discs wrt s vanishes (this is the first resultant)

res:=factor(discrim(discs,s)):nops(res);

IiIl

n:=10: for i from 1 to nops(res) do series(subs(F0=F0ser(n),S1=coeff(Ssser(n),x,1),op(i,res)),t,n+1); od;

KyVJInRHNiIiKDsnejsiIiE=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KzFJInRHNiIhIiIiIiEiI20iIiMhJWA7IiIlIiZXKT0iIichJnFqKSIiKSImSUslIiM1LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNw==

KyVJInRHNiIiIiIiIic=

e3:=op(2,res):

degree(discs,s);

IiIl

We now construct the next (generalized) resultant (see the paper MBM+Jehanne for details)

with(LinearAlgebra):

M:=Matrix(1..5,1..5);

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

for j from 1 to 5 do M[1,j]:=coeff(discs,s,5-j): od:

for j from 2 to 5 do M[2,j]:=coeff(discs,s,6-j): od:

for j from 1 to 4 do M[3,j]:=coeff(diff(discs,s),s,4-j): od:

for j from 2 to 5 do M[4,j]:=coeff(diff(discs,s),s,5-j): od:

for j from 3 to 5 do M[5,j]:=coeff(diff(discs,s),s,6-j): od:

The determinant of M mus also vanish

res:=factor(Determinant(M)): nops(res);

IiIl

n:=10: for i from 1 to nops(res) do series(subs(F0=F0ser(n),S1=coeff(Ssser(n),x,1),op(i,res)),t,n+1); od;

KyVJInRHNiIiJ2MpSCciIiE=

KyVJInRHNiIiIiIiIzU=

KzFJInRHNiIhIiIiIiEhIiUiIiMhIzciIiUhI2ciIichJFclIiIpISUlKVIiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

e4:=op(4,res):

Now eliminate F0

res:=factor(resultant(e3,e4,F0)): nops(res);

IiM2

n:=10: for i from 1 to nops(res) do series(subs(F0=F0ser(n),S1=coeff(Ssser(n),x,1),op(i,res)),t,n+1); od;

KyVJInRHNiIiSTdkQkxcIkg0WVpoIlFVclorcnNxQSIiIQ==

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

KzFJInRHNiIhIiIiIiEiIiQiIiMiIzciIiUiIyMpIiInIiQ5KCIiKSIlOHIiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiRik=

KzFJInRHNiIiIiIiIiEhI2EiIiMiJFgqIiIlISVHVyIiJyEmR3oiIiIpISYnM0EiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

KzFJInRHNiIiJ1Q5YCIiISEqR0gkUTUiIiMiK18zTzUkKiIiJSEtTy5BNm5dIiInIi81cFRAa2s9IiIpITBzPSQpKXohKikpWyIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

KzFJInRHNiIhI0YiIiEiJSEzIiIiIyEmOEEiIiIlIiYnKWUiIiInIidmWTciIikiJ1dkeiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

KzFJInRHNiIiIiIiIiEhIyEqIiIjIiVITSIiJSEmKVFyIiInIid5KHApIiIpISgxXzMnIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

KzFJInRHNiIiJGMjIiIhIiRvKCIiIyEkcyciIiUhJU8qKiIiJyEmZlAnIiIpISdPMF8iIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KzFJInRHNiIiIiIiIiEiI3MiIiMiJTdBIiIlIiZrJ1EiIiciJ3FYVyIiKSIoJ0gqKVEiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

KzFJInRHNiIiIiIiIiEhJDMiIiIjIiVvXCIiJSEnP2U3IiInIig7Wyk9IiIpISlzOV47IiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

op(2,res);

KiQpSSJ0RzYiIiNDIiIi

op(5,res);

KiQpLDwqKCIkSCgiIiIpSSNTMUc2IiIiJUYnKUkidEdGKiIiJ0YnRicqKCIkcypGJylGKSIiJEYnRixGJ0YnKigiJHkkRicpRikiIiNGJ0YsRichIiIqKCIkViNGJ0YxRicpRi1GK0YnRicqKCIkISkqRidGKUYnRixGJ0Y3KigiJDgmRidGNUYnRjpGJ0Y3KiYiJFYkRidGLEYnRjcqKCIkKltGJ0YpRidGOkYnRicqKCIjRkYnRjVGJylGLUY2RidGJyomIiNQRidGOkYnRicqKCIjVUYnRilGJ0ZFRidGNyokRkVGJ0Y3RilGJ0Y2Ric=

algS1:=collect(op(1,op(5,res)),S1,factor);

An equation for F0

res:=factor(resultant(e3,algS1,S1)): nops(res);

IiIk

n:=10: for i from 1 to nops(res) do series(subs(F0=F0ser(n),op(i,res)),t,n+1); od;

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNV

KzFJInRHNiIiOilbVFZgZk4hR2ojNHMhKiIiISE8XyRbIjQxISlbOyg+IXA8QiYiIiMhPmc8Klw/bXRPI1EkSG5TNlciIiUiPytXPEwnUkFaJClcJil6JFwvWCIiJyJCOyEzYiV5T3lhWD5FOD0jeSVwKSIiKSFFJWUmNFxSRT0uYHppcGlRVSdRMCIiIzUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

op(1,res);

KiQpSSJ0RzYiIiNVIiIi

op(3,res);

KiQpLD4qJiInIVsxIiIiIilJInRHNiIiIilGJyEiIiooIidLKzdGJ0kjRjBHRipGJylGKSIiJ0YnRiwqKCImI1JTRicpRi8iIiNGJylGKSIiJUYnRiwqJiInL2xvRidGMEYnRiwqKCIlVz5GJylGLyIiJEYnKUYpRjVGJ0YsKigiJ3dXT0YnRi9GJ0Y2RidGJyomIiRIKEYnKUYvRjdGJ0YnKigiJlkvJUYnRjRGJ0Y+RidGLComIiZDRiZGJ0Y2RidGJyomIiUsPEYnRjxGJ0YnKigiJldCI0YnRi9GJ0Y+RidGLComIiVCOEYnRjRGJ0YnKiYiJUg1RidGPkYnRiwqJiIkViRGJ0YvRidGJ0Y1Ric=

algF0:=collect(op(1,op(3,res)),F0);

LDIqJiIkSCgiIiIpSSNGMEc2IiIiJUYlRiUqJiwmKiYiJVc+RiUpSSJ0R0YoIiIjRiUhIiIiJSw8RiVGJSlGJyIiJEYlRiUqJiwoKiYiJiNSU0YlKUYvRilGJUYxKiYiJlkvJUYlRi5GJUYxIiVCOEYlRiUpRidGMEYlRiUqJiwqKiYiJ0srN0YlKUYvIiInRiVGMSomIid3V09GJUY5RiVGJSomIiZXQiNGJUYuRiVGMSIkViRGJUYlRidGJUYlKiYiJyFbMSJGJSlGLyIiKUYlRjEqJiInL2xvRiVGQkYlRjEqJiImQ0YmRiVGOUYlRiUqJiIlSDVGJUYuRiVGMQ==

The equations for S1 and P0

algS1:=729*S1^4*t^6+243*t^4*(4*t^2+1)*S1^3-27*t^2*(14*t^4+19*t^2-1)*S1^2-(20*t^2-1)*(7*t^2-6*t+1)*(7*t^2+6*t+1)*S1-t^2*(343*t^4-37*t^2+1);

algF0:=729*F0^4+(-1944*t^2+1701)*F0^3+(-40392*t^4-40446*t^2+1323)*F0^2+(-120032*t^6+364476*t^4-22344*t^2+343)*F0-106480*t^8-686504*t^6+52724*t^4-1029*t^2;

n:=22: series(subs(S1=coeff(Ssser(n),x,1),algS1),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

series(subs(F0=F0ser(n),algF0),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

<Text-field style="Heading 2" layout="Heading 2">Parametrization for F0 and S1</Text-field>

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

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

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpRigiIiRGKiksJkYoRiVGLEYlRixGJUYl

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

Tmser(5);Tmser(6);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

Expression of F0

res:=numer(factor(subs(t=sqrt(t2Tm),algF0)));

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

n:=6:normal(series(subs(Tm=Tmser(n),F0=F0ser(n),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),F0=F0ser(n),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiKk9KSz4oIiIhISs7KVFmSiUiIiMhLCVRJ0hMJ1wiIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

F0Tm:=factor(solve(op(1,res),F0));

LCQqKiMiIiIiJEciRiUsJkkjVG1HNiJGJUYlISIiRiUsKiomIiImRiUpRigiIiRGJUYlKiYiIzZGJSlGKCIiI0YlRioqJiIkTiJGJUYoRiVGJSIjTEYqRiVGLkYqRiU=

Expression of S1

res:=numer(factor(subs(t=sqrt(t2Tm),algS1)));

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

n:=6:normal(series(subs(Tm=Tmser(n),S1=coeff(Ssser(n),x,1),op(1,res)),t,n+1)); normal(series(subs(Tm=Tmser(n),S1=coeff(Ssser(n),x,1),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiJ1dARSIiISEoJ0hmQiIiIyEpb1YoKT0iIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

S1Tm:=factor(solve(op(1,res),S1));

KigsJkkjVG1HNiIiIiJGJiEiIkYmLCgqJClGJCIiI0YmRiYqJiIiJ0YmRiRGJkYnIiNARiZGJiksJkYkRiYiIiRGJkYxRic=

Results of this subsection

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

LCQqKEkjVHRHNiIiIiIsJkYkRiZGJiEiIkYoKSwmRiZGJiomIiIkRiZGJEYmRiZGLEYoRig=

LCQqKiMiIiIiJGMjRiUsJkkjVG1HNiJGJUYlISIiRiUpLCZGKEYlIiIkRiVGLUYlKUYoRi1GKkYl

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

Tmser(5);

KytJInRHNiIiIiIiIiEiIiUiIiMiI09GJy1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJg==

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

LCwqKCIkYyMiIiIpSSNUbUc2IiIiJEYlKUkidEdGKCIiI0YlRiUqJClGJyIiJUYlISIiKiYiIilGJUYmRiVGMComIiM9RiUpRidGLEYlRjAiI0ZGJQ==

F0Tm:=(1/128)*(Tm-1)*(5*Tm^3-11*Tm^2+135*Tm-33)/Tm^3;

LCQqKiMiIiIiJEciRiUsJkkjVG1HNiJGJUYlISIiRiUsKiomIiImRiUpRigiIiRGJUYlKiYiIzZGJSlGKCIiI0YlRioqJiIkTiJGJUYoRiVGJSIjTEYqRiVGLkYqRiU=

S1Tm:=(Tm-1)*(Tm^2-6*Tm+21)/(Tm+3)^3;

KigsJkkjVG1HNiIiIiJGJiEiIkYmLCgqJClGJCIiI0YmRiYqJiIiJ0YmRiRGJkYnIiNARiZGJiksJkYkRiYiIiRGJkYxRic=

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

eqS;

Recall that Ss is even in t

Ssser(6);

LCgqKEkieEc2IiIiIiwoKiYiIzVGJilGJCIiI0YmRiYqJiIjT0YmRiRGJkYmIiNQRiZGJilJInRHRiUiIidGJkYmKihGJEYmLCYqJiIiJEYmRiRGJkYmIiImRiZGJilGMCIiJUYmRiYqJilGMEYrRiZGJEYmRiY=

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

algSsTm0:=collect(numer(factor(subs(F0=F0Tm,S1=S1Tm,eqS))),Ssx,factor);

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

We express this in terms of Tm only, using the expression of t^2 in terms of Tm

algSsTm:=collect(op(3,factor(subs(t=sqrt(t2Tm),algSsTm0))),Ssx,factor);

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

n:=10: normal(series(subs(Tm=Tmser(n),Ssx=Ssser(n),algSsTm),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

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

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

IiQ7Jg==

degree(res,Ssx);

IiM3

Parametrisation of x (from the Gessel paper)

xTmVm:=(1-Tm+3*Vm+Vm*Tm)/Vm^2/(3+Vm+Tm-Vm*Tm);

KigsKiomSSNUbUc2IiIiIkkjVm1HRiZGJ0YnRiUhIiIqJiIiJEYnRihGJ0YnRidGJ0YnKUYoIiIjRiksKkYkRilGJUYnRihGJ0YrRidGKQ==

Expansion of Vm in terms of (Tm-1)

with(gfun): VmserTm:=proc(n) series(Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Vm=Tm1*VmsTm1,numer(x-xTmVm)), Tm1,VmsTm1,n,true)),Tm1,n+2); end:

VmserTm(6);

KzFJJFRtMUc2IiMiIiIiIiVGJiwmKiYjRiYiIztGJkkieEdGJEYmRiZGKiEiIiIiIywoKiYjRiYiI0tGJilGLEYuRiZGJiomRjFGJkYsRiZGLSNGJiIja0YmIiIkLCoqJiNGJiIkRyJGJkYsRiZGJiomI0Y3RjtGJkYzRiZGLSomIyIiJiIkYyNGJilGLEY3RiZGJiNGJkZBRi1GJywqKiYjIiIoIiVDNUYmRjNGJkYmRj5GLSomI0ZHIiQ3JkYmKUYsRidGJkYmI0YmRkhGJkZALC4qJiMiI0giJSc0JUYmRkJGJkYmKiYjIiNOIiVbP0YmRkxGJkYtKiYjRkBGUkYmRixGJkYtKiYjIiNARlZGJilGLEZARiZGJiNGJkZSRi0qJkZYRiZGM0YmRiYiIictSSJPRyUqcHJvdGVjdGVkRzYjRiZGRw==

Let us check

n:=9:normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),numer(x-xTmVm)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

Let us prove that Vm parametrises the solution

res:=numer(factor(subs(x=xTmVm,algSsTm))):nops(%);indets(%%);

IiIj

PCVJJFNzeEc2IkkjVG1HRiRJI1ZtR0Yk

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

IiIi

IiIj

The quadratic term does not vanish

n:=10: normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Ssx=Ssser(n),op(1,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=14: normal(series(subs(Vm=VmserTm(n),Tm1=Tmser(n)-1,Tm=Tmser(n),Ssx=Ssser(n),op(2,res)),t,n+1));

KylJInRHNiIsJiomIiVDNSIiIilJInhHRiQiIiNGKCEiIiomIiVbP0YoRipGKEYoIiM1LCoqJiIlJzQlRigpRioiIiRGKEYsKiYiJjNJJUYoRilGKEYsKiYiJmNpKkYoRipGKEYoIiU7IypGLCIjNy1JIk9HJSpwcm90ZWN0ZWRHNiNGKCIjOQ==

SsTmVm:=factor(solve(op(1,res),Ssx));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJI1ZtR0YqRiVGJUYpISIiKiYiIiRGJUYrRiVGJUYlRiVGJSwuKihGLkYlRilGJSlGK0YmRiVGJSooRiZGJUYpRiVGK0YlRiwqJkYuRiVGMUYlRixGKUYlKiYiIidGJUYrRiVGLEYuRiVGJSwmRitGJUYlRiVGLCwyKiYpRilGJkYlRjFGJUYlKihGJkYlRjlGJUYrRiVGLCooRiZGJUYpRiVGMUYlRiUqJEY5RiVGJUYzRiwqJkYmRiVGKUYlRiVGNEYsRi5GLEYsRiU=

Let's check

n:=18: normal(series(subs(Vm=convert(VmserTm(n),polynom),Tm=1+Tm1,Tm1=convert(Tmser(n)-1,polynom),SsTmVm),t,n+1));

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

Ssser(16);

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

normal(series(%%-%,t,17));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM9

Expression of txN(0,x) (Vmd is Vm for x^2 rather than x)

N0xftxTmVm:=subs(Vm=Vmd,SsTmVm);

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsLiooRi5GJUYpRiUpRitGJkYlRiUqKEYmRiVGKUYlRitGJUYsKiZGLkYlRjFGJUYsRilGJSomIiInRiVGK0YlRixGLkYlRiUsJkYrRiVGJUYlRiwsMiomKUYpRiZGJUYxRiVGJSooRiZGJUY5RiVGK0YlRiwqKEYmRiVGKUYlRjFGJUYlKiRGOUYlRiVGM0YsKiZGJkYlRilGJUYlRjRGLEYuRixGLEYl

let us check

n:=12: normal(series(subs(Vmd=subs(x=x^2,VmserTm(n)),Tm1=Tm-1,Tm=Tmser(n),N0xftxTmVm),t,n+1));

Ky1JInRHNiIqJClJInhHRiQiIiMiIiJGKCwmKiYiIiRGKSlGJyIiJUYpRikqJiIiJkYpRiZGKUYpRi4sKComIiM1RikpRiciIidGKUYpKiYiI09GKUYtRilGKSomIiNQRilGJkYpRilGNSwqKiYiI05GKSlGJyIiKUYpRikqJiIkJj5GKUY0RilGKSomIiQyJUYpRi1GKUYpKiYiJEskRilGJkYpRilGPi1JIk9HJSpwcm90ZWN0ZWRHNiNGKUYz

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

The cubic equation for Ss(x) over Q(x,T)

algSsTm:=(4*(x+1))*(Tm^4*x^2+2*Tm^4*x+8*Tm^3*x^2+Tm^4-48*Tm^3*x+18*Tm^2*x^2+8*Tm^3+36*Tm^2*x+18*Tm^2-27*x^2-54*x-27)*Ssx^3+(-12*Tm^4*x^2-24*Tm^4*x-96*Tm^3*x^2-12*Tm^4+576*Tm^3*x-216*Tm^2*x^2-96*Tm^3-432*Tm^2*x-216*Tm^2+324*x^2+648*x+324)*Ssx^2+(-3*Tm^4*x^3+7*Tm^4*x^2-24*Tm^3*x^3+19*Tm^4*x+168*Tm^3*x^2-54*Tm^2*x^3+9*Tm^4-504*Tm^3*x+270*Tm^2*x^2+72*Tm^3+486*Tm^2*x-432*Tm*x^2+81*x^3+162*Tm^2-432*Tm*x+243*x^2-81*x-243)*Ssx-(Tm-1)*(Tm^3*x^2-2*Tm^3*x+9*Tm^2*x^2+9*Tm^3-18*Tm^2*x+27*Tm*x^2-63*Tm^2-54*Tm*x+27*x^2+243*Tm-54*x-189)*x;

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

Parametrisation of x (from the Gessel paper)

xTmVm:=(1-Tm+3*Vm+Vm*Tm)/Vm^2/(3+Vm+Tm-Vm*Tm);

KigsKiomSSNUbUc2IiIiIkkjVm1HRiZGJ0YnRiUhIiIqJiIiJEYnRihGJ0YnRidGJ0YnKUYoIiIjRiksKkYkRilGJUYnRihGJ0YrRidGKQ==

Expansion of Vm in terms of (Tm-1)

with(gfun): VmserTm:=proc(n) series(Tm1*op(algeqtoseries(subs(Tm=1+Tm1,Vm=Tm1*VmsTm1,numer(x-xTmVm)), Tm1,VmsTm1,n,true)),Tm1,n+2); end:

N0xftxTmVmd := (1/2)*(Tm*Vmd-Tm+3*Vmd+1)*(3*Tm*Vmd^2-2*Tm*Vmd-3*Vmd^2+Tm-6*Vmd+3)/((Tm^2*Vmd^2-2*Tm^2*Vmd+2*Tm*Vmd^2+Tm^2-3*Vmd^2+2*Tm-6*Vmd-3)*(Vmd+1));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsLiooRi5GJUYpRiUpRitGJkYlRiUqKEYmRiVGKUYlRitGJUYsKiZGLkYlRjFGJUYsRilGJSomIiInRiVGK0YlRixGLkYlRiUsMiomKUYpRiZGJUYxRiVGJSooRiZGJUY4RiVGK0YlRiwqKEYmRiVGKUYlRjFGJUYlKiRGOEYlRiVGM0YsKiZGJkYlRilGJUYlRjRGLEYuRixGLCwmRitGJUYlRiVGLEYl

SsTmVm := (1/2)*(Tm*Vm-Tm+3*Vm+1)*(3*Tm*Vm^2-2*Tm*Vm-3*Vm^2+Tm-6*Vm+3)/((Tm^2*Vm^2-2*Tm^2*Vm+2*Tm*Vm^2+Tm^2-3*Vm^2+2*Tm-6*Vm-3)*(Vm+1));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJI1ZtR0YqRiVGJUYpISIiKiYiIiRGJUYrRiVGJUYlRiVGJSwuKihGLkYlRilGJSlGK0YmRiVGJSooRiZGJUYpRiVGK0YlRiwqJkYuRiVGMUYlRixGKUYlKiYiIidGJUYrRiVGLEYuRiVGJSwyKiYpRilGJkYlRjFGJUYlKihGJkYlRjhGJUYrRiVGLCooRiZGJUYpRiVGMUYlRiUqJEY4RiVGJUYzRiwqJkYmRiVGKUYlRiVGNEYsRi5GLEYsLCZGK0YlRiVGJUYsRiU=

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

map(factor,f13);

LCoqKCwuKiYtSSJTRzYiNiNJInhHRigiIiIpRioiIiNGK0YrKihGLUYrLUYnNiMqJEYqISIiRitGLEYrRjIqJkYtRitGLEYrRitGJkYrKiZGLUYrRi9GK0YyRitGMkYrKUkjRGVHRigjRitGLUYrLCYqJEYsRitGK0YrRitGMkYrKigiIiRGK0Y4RistSSJSR0YoRilGK0YrSSNSMEdGKEYyKiYsJkYzRitGK0YyRitGOEYyRjI=

f13s:=collect(map(expand,subs({De=Des,S(x)=Ss(x),S(1/x)=Ss(1/x),R(x)=Rs(x),x=sqrt(x),R0=Rs0},factor(f13))),[Des,Ss,Rs,Rs0],factor);

LCoqJiwoLUkjU3NHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwKiYsJiomRixGKkYpRipGKkYqRjBGKiwmRilGKkYqRipGMEYqRiopSSREZXNHRicjRipGLEYqRioqJiwmKiYiIiRGKkYpRipGKkY7RipGKi1JI1JzR0YnRihGKkYqSSRSczBHRidGMEYxRjA=

We can express Rs0 in terms of Ss(-1)

Rs0sol;

LCQtSSNTc0c2IjYjISIiRic=

At x=-1, the equation satisfied by Ss becomes quadratic (instead of cubic)

factor(subs(x=-1,algSsTm));

LDAqKCIkbygiIiIpSSRTc3hHNiIiIiNGJSlJI1RtR0YoIiIkRiUhIiIqKEYkRiVGJ0YlRipGJUYlKiYiIzdGJSlGKyIiJUYlRiUqJiIjW0YlRipGJUYtKiYiJGckRiUpRitGKUYlRiUqJiIkSyVGJUYrRiVGLSIkMyJGJQ==

factor(subs(Tm=Zm^2,%));

LCQqKCIjNyIiIiwsKigiIilGJUkkU3N4RzYiRiUpSSNabUdGKiIiJEYlRiUqJClGLCIiJUYlRiUqJkYwRiVGK0YlISIiKiYiIidGJSlGLCIiI0YlRiVGLUYyRiUsLEYnRiVGLkYyRjFGMkYzRjJGLUYlRiVGMg==

Ssm1solens:=factor([solve(%,Ssx)]);

NyQsJCoqIyIiIiIiKUYmLCZJI1ptRzYiRiZGJkYmRiYsKiokKUYpIiIkRiZGJiomRi5GJilGKSIiI0YmRiYqJkYuRiZGKUYmRiZGLiEiIkYmRi1GM0YmLCQqKkYlRiYsJkYpRiZGJkYzRiYsKkYsRiZGL0YzRjJGJkYuRiZGJkYtRjNGMw==

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

Zmser(10);

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpLUkiT0clKnByb3RlY3RlZEc2I0YlIiM1

subs(x=-1,Ssser(5));

LCgqJiIjNiIiIilJInRHNiIiIidGJSEiIiomIiIjRiUpRiciIiVGJUYqKiQpRidGLEYlRio=

n:=10: normal(series(subs(Zm=Zmser(n), Ssm1solens[1]),t,n+1)); normal(series(subs(Zm=Zmser(n), Ssm1solens[2]),t,n+1));

Ky9JInRHNiIiIiIiIiFGJSIiI0YnIiIlIiM2IiInIiMmKSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

Ky1JInRHNiIhIiIiIiMhIiMiIiUhIzYiIichIyYpIiIpLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNQ==

Ssm1Zm:=Ssm1solens[2];

LCQqKiMiIiIiIilGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKiomRi5GJUYoRiVGJUYuRiVGJUYtRipGKg==

Rs0Zm:=-Ssm1Zm;

LCQqKiMiIiIiIilGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKiomRi5GJUYoRiVGJUYuRiVGJUYtRipGJQ==

subs(x=0,Rsser(5));

LCgqJiIjNiIiIilJInRHNiIiIidGJUYlKiYiIiNGJSlGJyIiJUYlRiUqJClGJ0YrRiVGJQ==

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

f13s;

res:=numer(factor(resultant(subs(Ssx=DD+2*Ss1x-(2*x-1)/(1+x),algSsTm),subs(Ssx=Ss1x,x=1/x,algSsTm),Ss1x))): nops(res);

IiIl

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

NiYiIiIiIiMiI3ciJEom

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

NiYiIiFGIyIiJCIiJw==

n:=1: normal(series(subs(DD=Ssser(n)-2*subs(x=1/x,Ssser(n))+(2*x-1)/(1+x),Tm=Tmser(n),op(3,res)),t,n+1));

KydJInRHNiIsJCooIiQ3JiIiIkkieEdGJEYoLCoqJiIiJkYoKUYpIiIkRihGKComIiInRigpRikiIiNGKEYoKiZGLkYoRilGKCEiIiIiJUY0RihGKCIiIS1JIk9HJSpwcm90ZWN0ZWRHNiNGKEYo

n:=10: normal(series(subs(DD=Ssser(n)-2*subs(x=1/x,Ssser(n))+(2*x-1)/(1+x),Tm=Tmser(n),op(4,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

indets(op(4,res));

PCVJI0RERzYiSSNUbUdGJEkieEdGJA==

algDDTm:=op(4,res):

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);

IiV4WQ==

degree(res,DD);

IiND

Back to Rs

f13s;

res:=numer(factor(subs(DD=(-3*(x+1)*Rsx+Rs0Zm+(2*x-1)/(1+x))/sqrt(Deltas),Tm=Zm^2,algDDTm))): nops(%);

IiIl

op(1,res);nops(op(2,res));nops(op(3,res));nops(op(4,res));

ISIi

IiIj

IiVRaA==

IiIi

op(1,res); op(2,res);op(4,res);

ISIi

LCZJInhHNiIiIiJGJUYl

SSJ4RzYi

degree(op(3,res),Rsx);degree(op(3,res),t);

IiIn

indets(op(3,res));

PCZJJFJzeEc2IkkjWm1HRiRJInRHRiRJInhHRiQ=

res1:=numer(factor(subs(t=sqrt(t2Tm),Tm=Zm^2,op(3,res)))): nops(res1);

IiIl

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

NiYiIiIiIiMiIyUqIiQ9Ig==

NiY8IjwkSSNabUc2IkkieEdGJjwlSSRSc3hHRiZGJUYnRig=

NiYiIiFGIyIiJEYk

n:=10: normal(series(subs(Zm=Zmser(n),Rsx=Rsser(n),op(3,res1)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=10: normal(series(subs(Zm=Zmser(n),Rsx=Rsser(n),op(4,res1)),t,n+1));

Ky9JInRHNiIsKComIiUjPikiIiIpSSJ4R0YkIiIkRighIiIqJiImIVs/RigpRioiIiNGKEYoKiZGJ0YoRipGKEYsIiIhLCwqJiIma28kRigpRioiIiVGKEYoKiYiJzNtPkYoRilGKEYsKiYiJ08uRkYoRi9GKEYoKiZGOUYoRipGKEYsRjVGKEYwLCwqJiInIUdQKEYoRjZGKEYoKiYiKCs/MiRGKEYpRihGLComIihLR1olRihGL0YoRigqJkZBRihGKkYoRixGP0YoRjcsLComIikjUiI0N0YoRjZGKEYoKiYiKXNZa1pGKEYpRihGLComIilLKTMrKEYoRi9GKEYoKiZGSUYoRipGKEYsRkdGKCIiJywsKiYiKmc0Wyo9RihGNkYoRigqJiIqVzwkR3RGKEYpRihGLComIislKT5zejVGKEYvRihGKComRlJGKEYqRihGLEZQRigiIiktSSJPRyUqcHJvdGVjdGVkRzYjRigiIzU=

Equation of R(x) over Q(x,Z)

algRsZm:=op(3,res1): degree(%,Rsx);

IiIk

Equation for Rs(x) over Q(x,t)

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

IiUmMyU=

degree(res,Rsx);

IiND

algRs:=res:

n:=10:normal(series(subs(Rsx=Rsser(n),algRs),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

Parametrization

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

IiIk

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

NiUiIiIiI1UiJFkk

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

NiU8IjwlSSRSc3hHNiJJI1ZtR0YmSSNabUdGJkYk

NiUiIiEiIiIiIiM=

n:=10: normal(series(subs(Vm=VmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rsx=Rsser(n),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

n:=22: normal(series(subs(Vm=VmserTm(n),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Rsx=Rsser(n),op(3,res)),t,n+1));

KylJInRHNiIsJiomIiZvRiQiIiIpSSJ4R0YkIiIkRihGKComRidGKClGKiIiI0YoISIiIiM/LCwqJiIndyRII0YoKUYqIiIlRihGKComIihTOWkjRihGKUYoRigqJiIoKSk9KUhGKEYtRihGLyomIiYvJCkqRihGKkYoRihGJ0YoIiNBLUkiT0clKnByb3RlY3RlZEc2I0YoIiND

Rational expression of Rs in terms of Zm and Vm

RsZmVm:=factor(solve(op(2,res),Rsx));

LCQqMCMiIiIiIztGJUkjVm1HNiJGJSwqKiZGJ0YlKUkjWm1HRigiIiNGJUYlKiRGK0YlISIiRidGLyIiJEYvRiUsSiomKUYnRjBGJSlGLCIiJUYlRiUqKEYtRiVGM0YlKUYsRjBGJUYvKiYpRidGLUYlRjRGJUYlKihGNUYlRjNGJUYrRiVGJSooRi1GJUY5RiVGN0YlRiUqJkYnRiVGNEYlRiUqKCIiJ0YlRjNGJUYsRiVGLyooIiIpRiVGOUYlRitGJUYvKihGLUYlRidGJUY3RiVGLyokRjRGJUYlKiZGMEYlRjNGJUYlKihGLUYlRjlGJUYsRiVGLyooRkBGJUYnRiVGK0YlRi8qJkYtRiVGN0YlRiUqJiIiKEYlRjlGJUYlKihGLUYlRidGJUYsRiVGJSomRjVGJUYrRiVGJSomRkhGJUYnRiVGJSomRj5GJUYsRiVGJUYwRiVGJUY3Ri8pLCZGJ0YlRiVGJUYtRi8pLCoqJkYnRiVGLEYlRiVGJ0YvRixGL0YlRi9GLUYvRi8=

Back to N(x,0)

op(Rsser); op(Rser);

Zio2I0kibkc2IkYlRiVGJS1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ4R0YlLUklc3FydEc2JEYoSShfc3lzbGliR0YlNiNGKy1JJVJzZXJHRiVGI0YlRiVGJQ==

Zio2I0kibkc2IkYlRiVGJS1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSooKUkidEdGJSIiIyIiIkkieEdGJSEiIi1JJXN1YnNHRik2JC9JInlHRiUiIiEtSSVOc2VyR0YlNiMsJkYkRjBGMEYyRjBGLkknZmFjdG9yR0YoRiVGJUYl

Expression of N(x,0)/x (Vmd is Vm evaluated at x^2 rather than x)

Nx0sxZmVmd:=factor(subs(Vm=Vmd,RsZmVm)/subs(Tm=Zm^2,t2Tm));

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

Let us check

n:=12: normal(series(subs(Vmd=subs(x=x^2,VmserTm(n)),Tm1=Tm-1,Tm=Zm^2,Zm=Zmser(n),Nx0sxZmVm),t,n+1));

Ky9JInRHNiIiIiIiIiEsJiokKUkieEdGJCIiI0YlRiVGK0YlRissKComRitGJSlGKiIiJUYlRiUqJiIiKUYlRilGJUYlIiM2RiVGLywqKiYiIiZGJSlGKiIiJ0YlRiUqJiIjSUYlRi5GJUYlKiYiI3RGJUYpRiVGJSIjJilGJUY3LCwqJiIjOUYlKUYqRjFGJUYlKiYiJDciRiVGNkYlRiUqJiIkKFFGJUYuRiVGJSomIiRSKEYlRilGJUYlIiQjeUYlRjEtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzU=

subs(y=0,Nser(8)/x);

KiYsLEkieEc2IiIiIiomLCYqJClGJCIiJEYmRiYqJiIiI0YmRiRGJkYmRiYpSSJ0R0YlRi1GJkYmKiYsKComRi1GJilGJCIiJkYmRiYqJiIiKUYmRipGJkYmKiYiIzZGJkYkRiZGJkYmKUYvIiIlRiZGJiomLCoqJkY0RiYpRiQiIihGJkYmKiYiI0lGJkYzRiZGJiomIiN0RiZGKkYmRiYqJiIjJilGJkYkRiZGJkYmKUYvIiInRiZGJiomLCwqJiIjOUYmKUYkIiIqRiZGJiomIiQ3IkYmRj5GJkYmKiYiJChRRiZGM0YmRiYqJiIkUihGJkYqRiZGJiomIiQjeUYmRiRGJkYmRiYpRi9GNkYmRiZGJkYkISIi

normal(series(%%-%,t,9));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

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

Rs0Zm:=(1/8)*(Zm-1)*(Zm^3-3*Zm^2+3*Zm+3)/Zm^3 ;

LCQqKiMiIiIiIilGJSwmSSNabUc2IkYlRiUhIiJGJSwqKiQpRigiIiRGJUYlKiZGLkYlKUYoIiIjRiVGKiomRi5GJUYoRiVGJUYuRiVGJUYtRipGJQ==

RsZmVm:=-(1/16)*Vm*(Vm*Zm^2-Zm^2-Vm-3)*(Vm^3*Zm^4-2*Vm^3*Zm^3+Vm^2*Zm^4+4*Vm^3*Zm^2+2*Vm^2*Zm^3+Vm*Zm^4-6*Vm^3*Zm-8*Vm^2*Zm^2-2*Vm*Zm^3+Zm^4+3*Vm^3-2*Vm^2*Zm-8*Vm*Zm^2+2*Zm^3+7*Vm^2+2*Vm*Zm+4*Zm^2+7*Vm+6*Zm+3)/(Zm^3*(Vm+1)^2*(Vm*Zm-Vm-Zm-1)^2) ;

Nx0sxZmVmd := -16*Vmd*(Vmd*Zm^2-Zm^2-Vmd-3)*(Vmd^3*Zm^4-2*Vmd^3*Zm^3+Vmd^2*Zm^4+4*Vmd^3*Zm^2+2*Vmd^2*Zm^3+Vmd*Zm^4-6*Vmd^3*Zm-8*Vmd^2*Zm^2-2*Vmd*Zm^3+Zm^4+3*Vmd^3-2*Vmd^2*Zm-8*Vmd*Zm^2+2*Zm^3+7*Vmd^2+2*Vmd*Zm+4*Zm^2+7*Vmd+6*Zm+3)*Zm^3/((Vmd+1)^2*(Vmd*Zm-Vmd-Zm-1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

We import results obtained earlier for the series M (when studying walks starting at (0,0))

Mx0sxZmVmd := -(16/3)*(Vmd*Zm^2-Zm^2-Vmd-3)*Vmd*(Vmd^3*Zm^4-2*Vmd^3*Zm^3-3*Vmd^2*Zm^4+4*Vmd^3*Zm^2-6*Vmd^2*Zm^3-3*Vmd*Zm^4-6*Vmd^3*Zm+8*Vmd^2*Zm^2+6*Vmd*Zm^3+Zm^4+3*Vmd^3-10*Vmd^2*Zm+8*Vmd*Zm^2+2*Zm^3+11*Vmd^2+10*Vmd*Zm+4*Zm^2+11*Vmd+6*Zm+3)*Zm^3/((Vmd+1)^2*(Vmd*Zm-Vmd-Zm-1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

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

M0xftxTmVmd := -(1/2)*(Tm*Vmd-Tm+3*Vmd+1)*(Tm*Vmd^2+2*Tm*Vmd-Vmd^2-Tm-2*Vmd-3)/((Vmd+1)*(Tm^2*Vmd^2-2*Tm^2*Vmd+2*Tm*Vmd^2+Tm^2-3*Vmd^2+2*Tm-6*Vmd-3));

LCQqLCMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsLiomRilGJSlGK0YmRiVGJSooRiZGJUYpRiVGK0YlRiUqJEYxRiVGLEYpRiwqJkYmRiVGK0YlRixGLkYsRiUsJkYrRiVGJUYlRiwsMiomKUYpRiZGJUYxRiVGJSooRiZGJUY4RiVGK0YlRiwqKEYmRiVGKUYlRjFGJUYlKiRGOEYlRiUqJkYuRiVGMUYlRiwqJkYmRiVGKUYlRiUqJiIiJ0YlRitGJUYsRi5GLEYsRiw=

The four specializations are odd in x or y

subs(x=0,Mser(4)); subs(y=0,Mser(4)); subs(x=0,Nser(4)); subs(y=0,Nser(4));

LCYqJkkieUc2IiIiIkkidEdGJUYmRiYqJiwmKiYiIiRGJilGJEYrRiZGJiomIiIoRiZGJEYmRiZGJilGJ0YrRiZGJg==

LCgqJiMiIiIiIiRGJUkieEc2IkYlRiUqJiwmKiYjIiIlRiZGJUYnRiVGJSomRiRGJSlGJ0YmRiVGJUYlKUkidEdGKCIiI0YlRiUqJiwoKiYjRjJGJkYlKUYnIiImRiVGJSomRi1GJUYvRiVGJSomIyIjSkYmRiVGJ0YlRiVGJSlGMUYtRiVGJQ==

LCYqJkkieUc2IiIiIkkidEdGJUYmRiYqJiwmKiYiIiRGJilGJEYrRiZGJiomIiImRiZGJEYmRiZGJilGJ0YrRiZGJg==

LChJInhHNiIiIiIqJiwmKiQpRiMiIiRGJUYlKiYiIiNGJUYjRiVGJUYlKUkidEdGJEYsRiVGJSomLCgqJkYsRiUpRiMiIiZGJUYlKiYiIilGJUYpRiVGJSomIiM2RiVGI0YlRiVGJSlGLiIiJUYlRiU=

<Text-field style="Heading 2" layout="Heading 2"><Font size="20">Back to the sections of L and B</Font></Text-field>

Lx0sxZmVmd:=factor((Mx0sxZmVmd+Nx0sxZmVmd)/2);

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

B0xsxZmVmd:=factor((Mx0sxZmVmd-Nx0sxZmVmd)/2);

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

let us check

n:=16: normal(series(subs(Vmd=VmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),x=x^2,Lx0sxZmVmd),t,n+1));

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

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

Ky9JInRHNiIjIiIjIiIkIiIhLCYjIiImRiciIiIqJkYlRiwpSSJ4R0YkRiZGLEYsRiYsKCMiI0tGJ0YsKiYiIidGLEYuRixGLComIyIiJUYnRiwpRi9GN0YsRixGNywqIyIkeSNGJ0YsKiYiI2dGLEYuRixGLComIyIjbEYnRixGOEYsRiwqJiMiIzVGJ0YsKUYvRjRGLEYsRjQsLComIyIjR0YnRiwpRi8iIilGLEYsKiYjIiQnKilGJ0YsRjhGLEYsKiYjIiRRI0YnRixGREYsRiwjIiU7R0YnRiwqJiMiJW0+RidGLEYuRixGLEZKLC4qJkZIRiwpRi9GQ0YsRiwqJiIkJUhGLEZJRixGLComIiUrOUYsRkRGLEYsKiYiJSYqUkYsRjhGLEYsKiYiJUJ3RixGLkYsRiwiJkgvIkYsRkM=

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

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

n:=16: normal(series(subs(Vmd=VmserTm(n),Tm1=Tm-1,Tm=Tmser(n),Zm=Zmser(n),x=x^2,B0xsxZmVmd),t,n+1));

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

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

Ky9JInRHNiIjISIiIiIkIiIhLCYqJiMiIiJGJ0YsKUkieEdGJCIiI0YsRiZGK0YmRi8sKEYrRiYqJkYvRixGLUYsRiYqJiNGL0YnRiwpRi4iIiVGLEYmRjUsKiMiI0JGJ0YsKiYiIzhGLEYtRixGJiomIyIjREYnRixGNEYsRiYqJiMiIiZGJ0YsKUYuIiInRixGJkZCLCwqJiMiIzlGJ0YsKUYuIiIpRixGJiMiJHElRidGLComIyIkbCNGJ0YsRjRGLEYmKiYjIiMpKkYnRixGQUYsRiYqJiMiJF4jRidGLEYtRixGJkZILC4qJkZGRiwpRi4iIzVGLEYmKiYiJEUiRixGR0YsRiYqJiIkdyVGLEZBRixGJiomIiQlKilGLEY0RixGJiomIiRIJUYsRi1GLEYmIiVEQ0YsRlc=

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

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

L0xftxTmVmd:=factor((M0xftxTmVmd+N0xftxTmVmd)/2);

LCQqLiMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsJkYrRiVGJUYsRiUsKkYoRiVGKUYsRitGLEYuRixGJSwmRitGJUYlRiVGLCwyKiYpRilGJkYlKUYrRiZGJUYlKihGJkYlRjRGJUYrRiVGLCooRiZGJUYpRiVGNUYlRiUqJEY0RiVGJSomRi5GJUY1RiVGLComRiZGJUYpRiVGJSomIiInRiVGK0YlRixGLkYsRixGJQ==

Bx0ftxTmVmd:=factor((M0xftxTmVmd-N0xftxTmVmd)/2);

LCQqLCwqKiZJI1RtRzYiIiIiSSRWbWRHRidGKEYoRiYhIiIqJiIiJEYoRilGKEYoRihGKEYoRilGKCwoRiVGKEYpRioiIiNGKkYoLCZGKUYoRihGKEYqLDIqJilGJkYuRigpRilGLkYoRigqKEYuRihGMkYoRilGKEYqKihGLkYoRiZGKEYzRihGKCokRjJGKEYoKiZGLEYoRjNGKEYqKiZGLkYoRiZGKEYoKiYiIidGKEYpRihGKkYsRipGKkYq

n:=16: normal(series(subs(Vmd=VmserTm(n),Tm1=Tm-1,Tm=Tmser(n),x=x^2,L0xftxTmVmd),t,n+1));

KzFJInRHNiIqJClJInhHRiQiIiMiIiJGKCwmKiYiIiRGKSlGJyIiJUYpRikqJiIiJ0YpRiZGKUYpRi4sKComIiM1RikpRidGMEYpRikqJiIjU0YpRi1GKUYpKiYiI15GKUYmRilGKUYwLCoqJiIjTkYpKUYnIiIpRilGKSomIiQ1I0YpRjRGKUYpKiYiJCVcRilGLUYpRikqJiIkNyZGKUYmRilGKUY9LCwqJiIkRSJGKSlGJ0YzRilGKSomIiUzNUYpRjxGKUYpKiYiJTxNRilGNEYpRikqJiIlb2hGKUYtRilGKSomIiVuY0YpRiZGKUYpRjMsLiomIiRpJUYpKUYnIiM3RilGKSomIiU/WUYpRkdGKUYpKiYiJiVRP0YpRjxGKUYpKiYiJiFSXkYpRjRGKUYpKiYiJmQoeUYpRi1GKUYpKiYiJndvJ0YpRiZGKUYpRlQtSSJPRyUqcHJvdGVjdGVkRzYjRikiIzk=

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

Ky1JInRHNiIqJClJInhHRiQiIiMiIiJGKCwkKigiIiRGKUYmRiksJkYlRilGKEYpRilGKSIiJSomRiZGKSwoKiYiIzVGKSlGJ0YuRilGKSomIiNTRilGJkYpRikiI15GKUYpIiInKiZGJkYpLCoqJiIjTkYpKUYnRjdGKUYpKiYiJDUjRilGM0YpRikqJiIkJVxGKUYmRilGKSIkNyZGKUYpIiIpLCQqKEYsRilGJkYpLCwqJiIjVUYpKUYnRkJGKUYpKiYiJE8kRilGPEYpRikqJiIlUjZGKUYzRilGKSomIiVjP0YpRiZGKUYpIiUqKT1GKUYpRilGMg==

normal(series(%%%-%,t,12));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

n:=16: normal(series(subs(Vmd=VmserTm(n),Tm1=Tm-1,Tm=Tmser(n),x=x^2,Bx0ftxTmVmd),t,n+1));

KzFJInRHNiIqJClJInhHRiQiIiMiIiIiIiUsJiomRipGKSlGJ0YqRilGKSomIiM5RilGJkYpRikiIicsKComIiM6RikpRidGMEYpRikqJiIjKClGKUYtRilGKSomIiQhPUYpRiZGKUYpIiIpLCoqJiIjY0YpKUYnRjlGKUYpKiYiJFclRilGNEYpRikqJiIlZzlGKUYtRilGKSomIiVDQkYpRiZGKUYpIiM1LCwqJiIkNSNGKSlGJ0ZERilGKSomIiUrQEYpRj1GKUYpKiYiJTAjKkYpRjRGKUYpKiYiJmlEI0YpRi1GKUYpKiYiJkUwJEYpRiZGKUYpIiM3LC4qJiIkI3pGKSlGJ0ZRRilGKSomIiVTJipGKUZIRilGKSomIiZrPSZGKUY9RilGKSomIiddZztGKUY0RilGKSomIid3dUxGKUYtRilGKSomIicnPTMlRilGJkYpRilGLy1JIk9HJSpwcm90ZWN0ZWRHNiNGKSIjOw==

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

KytJInRHNiIqJClJInhHRiQiIiMiIiIiIiUsJCooRihGKUYmRiksJiomRihGKUYmRilGKSIiKEYpRilGKSIiJywkKigiIiRGKUYmRiksKComIiImRikpRidGKkYpRikqJiIjSEYpRiZGKUYpIiNnRilGKUYpIiIpLCQqKEYqRilGJkYpLCoqJiIjOUYpKUYnRjBGKUYpKiYiJDYiRilGN0YpRikqJiIkbCRGKUYmRilGKSIkImVGKUYpRikiIzU=

normal(series(%%%-%,t,12));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM3

Algebraic equations for the sections

We start with L(x,0)/x, and build its cubic equation over Q(x,Zm)

Lx0sxZmVmd;

res:=factor(resultant(numer(Lx0sxZmVmd-Lx0sx),subs(Tm=Zm^2,numer(subs(Vm=Vmd,x^2-xTmVm))),Vmd)): nops(res);

IiIm

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

NiciIiIiIiNGJEYkIiQ9Iw==

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

Nic8IjwjSSNabUc2IkYkRiQ8JUkmTHgwc3hHRiZGJUkieEdGJg==

algLx0sxZm:=collect(op(5,res),Lx0sx,factor);

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

Now with txL(0,x) (over Q(x,Tm))

res:=factor(resultant(numer(L0xftxTmVmd-L0xftx),subs(Vm=Vmd,numer(x^2-xTmVm)),Vmd)): nops(res);

IiIl

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

NiYiIiIiIiNGJCIjYw==

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

NiY8IjwjSSNUbUc2IkYkPCVJJ0wweGZ0eEdGJkYlSSJ4R0Ym

algL0xftxTm:=collect(op(4,res),L0xftx,factor);

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

Now with txB(x,0) (again over Q(x,Tm))

Bx0ftxTmVmd;

res:=factor(resultant(numer(Bx0ftxTmVmd-Bx0ftx),subs(Vm=Vmd,numer(x^2-xTmVm)),Vmd)): nops(res);

IiIl

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

NiYiIiIiIiNGJCIjYA==

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

NiY8IjwjSSNUbUc2IkYkPCVJJ0J4MGZ0eEdGJkYlSSJ4R0Ym

algBx0ftxTm:=collect(op(4,res),Bx0ftx,factor);

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

Finally with B(0,x)/x

res:=factor(resultant(numer(B0xsxZmVmd-B0xsx),subs(Tm=Zm^2,Vm=Vmd,numer(x^2-xTmVm)),Vmd)): nops(res);

IiIm

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

NiciIiIiIiNGJEYkIiQ3Iw==

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

Nic8IjwjSSNabUc2IkYkRiQ8JUkmQjB4c3hHRiZGJUkieEdGJg==

algB0xsxZm:=collect(op(5,res),B0xsx,factor);

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

Degrees of these specializations over Q(t,x)

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

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

First, L(x,0)

res:=factor(resultant(algLx0sxZm,algZm,Zm)): nops(res);

IiUmNCU=

degree(res,Lx0sx);

IiND

Then L(0,x)

res:=factor(resultant(algL0xftxTm,algTm,Tm)): nops(res);

IiQqWw==

degree(res,L0xftx);

IiM3

Then B(x,0)

res:=factor(resultant(algBx0ftxTm,algTm,Tm)): nops(res);

IiQ2Jg==

degree(res,Bx0ftx);

IiM3

Finally B(0,x)

res:=factor(resultant(algB0xsxZm,algZm,Zm)): nops(res);

IiUmNCU=

degree(res,B0xsx);

IiND

xTmVm:=(Tm*Vm-Tm+3*Vm+1)/(Vm^2*(-Tm*Vm+Tm+Vm+3));

KigsKiomSSNUbUc2IiIiIkkjVm1HRiZGJ0YnRiUhIiIqJiIiJEYnRihGJ0YnRidGJ0YnKUYoIiIjRiksKkYkRilGJUYnRihGJ0YrRidGKQ==

Lx0sxZmVmd:=-(32/3)*(Vmd*Zm^2-Zm^2-Vmd-3)*Vmd*Zm^3*(Vmd^3*Zm^4-2*Vmd^3*Zm^3+4*Vmd^3*Zm^2-6*Vmd^3*Zm-4*Vmd^2*Zm^2+Zm^4+3*Vmd^3-4*Vmd^2*Zm-4*Vmd*Zm^2+2*Zm^3+8*Vmd^2+4*Vmd*Zm+4*Zm^2+8*Vmd+6*Zm+3)/((Vmd+1)^2*(Vmd*Zm-Vmd-Zm-1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

collect(-(Vmd^3*Zm^4-2*Vmd^3*Zm^3+4*Vmd^3*Zm^2-6*Vmd^3*Zm-4*Vmd^2*Zm^2+Zm^4+3*Vmd^3-4*Vmd^2*Zm-4*Vmd*Zm^2+2*Zm^3+8*Vmd^2+4*Vmd*Zm+4*Zm^2+8*Vmd+6*Zm+3),Vmd,factor);

LCoqKCwmKiQpSSNabUc2IiIiIyIiIkYqIiIkRipGKiksJkYnRipGKiEiIkYpRiopSSRWbWRHRihGK0YqRi4qKiIiJUYqLCZGJ0YqRilGKkYqRi1GKilGMEYpRipGKioqRjJGKiwmRidGKkYqRipGKiwmRidGKkYpRi5GKkYwRipGKiomRiRGKilGNkYpRipGLg==

L0xftxTmVmd:=(1/2)*(Tm*Vmd-Tm+3*Vmd+1)*(Vmd-1)*(Tm*Vmd-Tm-Vmd-3)/((Vmd+1)*(Tm^2*Vmd^2-2*Tm^2*Vmd+2*Tm*Vmd^2+Tm^2-3*Vmd^2+2*Tm-6*Vmd-3));

LCQqLiMiIiIiIiNGJSwqKiZJI1RtRzYiRiVJJFZtZEdGKkYlRiVGKSEiIiomIiIkRiVGK0YlRiVGJUYlRiUsJkYrRiVGJUYsRiUsKkYoRiVGKUYsRitGLEYuRixGJSwyKiYpRilGJkYlKUYrRiZGJUYlKihGJkYlRjNGJUYrRiVGLCooRiZGJUYpRiVGNEYlRiUqJEYzRiVGJSomRi5GJUY0RiVGLComRiZGJUYpRiVGJSomIiInRiVGK0YlRixGLkYsRiwsJkYrRiVGJUYlRixGJQ==

Bx0ftxTmVmd:=-(Tm*Vmd-Tm+3*Vmd+1)*Vmd*(Tm*Vmd-Vmd-2)/((Vmd+1)*(Tm^2*Vmd^2-2*Tm^2*Vmd+2*Tm*Vmd^2+Tm^2-3*Vmd^2+2*Tm-6*Vmd-3));

LCQqLCwqKiZJI1RtRzYiIiIiSSRWbWRHRidGKEYoRiYhIiIqJiIiJEYoRilGKEYoRihGKEYoRilGKCwoRiVGKEYpRioiIiNGKkYoLDIqJilGJkYuRigpRilGLkYoRigqKEYuRihGMUYoRilGKEYqKihGLkYoRiZGKEYyRihGKCokRjFGKEYoKiZGLEYoRjJGKEYqKiZGLkYoRiZGKEYoKiYiIidGKEYpRihGKkYsRipGKiwmRilGKEYoRihGKkYq

B0xsxZmVmd:=(16/3)*(Vmd*Zm^2-Zm^2-Vmd-3)*Vmd*Zm^3*(Vmd^3*Zm^4-2*Vmd^3*Zm^3+3*Vmd^2*Zm^4+4*Vmd^3*Zm^2+6*Vmd^2*Zm^3+3*Vmd*Zm^4-6*Vmd^3*Zm-16*Vmd^2*Zm^2-6*Vmd*Zm^3+Zm^4+3*Vmd^3+2*Vmd^2*Zm-16*Vmd*Zm^2+2*Zm^3+5*Vmd^2-2*Vmd*Zm+4*Zm^2+5*Vmd+6*Zm+3)/((Vmd+1)^2*(Vmd*Zm-Vmd-Zm-1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^3);

collect(Vmd^3*Zm^4-2*Vmd^3*Zm^3+3*Vmd^2*Zm^4+4*Vmd^3*Zm^2+6*Vmd^2*Zm^3+3*Vmd*Zm^4-6*Vmd^3*Zm-16*Vmd^2*Zm^2-6*Vmd*Zm^3+Zm^4+3*Vmd^3+2*Vmd^2*Zm-16*Vmd*Zm^2+2*Zm^3+5*Vmd^2-2*Vmd*Zm+4*Zm^2+5*Vmd+6*Zm+3, Vmd,factor);

LCoqKCwmKiQpSSNabUc2IiIiIyIiIkYqIiIkRipGKiksJkYnRipGKiEiIkYpRiopSSRWbWRHRihGK0YqRioqKCwoKiZGK0YqRiZGKkYqKiYiIzdGKkYnRipGKiIiJkYqRipGLEYqKUYwRilGKkYqKigsKEYzRipGNEYuRjZGKkYqKSwmRidGKkYqRipGKUYqRjBGKkYqKiZGJEYqRjpGKkYq

Now we can compute the GF of Gessel's walks ending on the diagonal (Section 6)

factor(Lx0sxZmVmd-B0xsxZmVmd);

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

collect(Vmd^3*Zm^4-2*Vmd^3*Zm^3+Vmd^2*Zm^4+4*Vmd^3*Zm^2+2*Vmd^2*Zm^3+Vmd*Zm^4-6*Vmd^3*Zm-8*Vmd^2*Zm^2-2*Vmd*Zm^3+Zm^4+3*Vmd^3-2*Vmd^2*Zm-8*Vmd*Zm^2+2*Zm^3+7*Vmd^2+2*Vmd*Zm+4*Zm^2+7*Vmd+6*Zm+3,Vmd,factor);

LCoqKCwmKiQpSSNabUc2IiIiIyIiIkYqIiIkRipGKiksJkYnRipGKiEiIkYpRiopSSRWbWRHRihGK0YqRioqKkYtRiosJkYnRipGKkYqRiosKEYlRioqJkYpRipGJ0YqRioiIihGLkYqKUYwRilGKkYqKipGLUYqRjJGKiwoRiVGKkY0Ri5GNUYuRipGMEYqRioqJkYkRiopRjJGKUYqRio=

The series in t occurring in the equations for L and B are the constant terms in L(x,0)/x and B(0,x)/x

solve(numer(xTmVm),Vm);

KiYsJkkjVG1HNiIiIiJGJiEiIkYmLCZGJEYmIiIkRiZGJw==

L10Zm:=factor(subs(Vmd=%,Tm=Zm^2,Lx0sxZmVmd)); B01Zm:=factor(subs(Vmd=%%,Tm=Zm^2,B0xsxZmVmd));

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

Let us check this

n:=10: series(subs(Zm=Zmser(n),L10Zm),t,n+1);

Ky9JInRHNiIjIiIjIiIkIiIhIyIiJkYnRiYjIiNLRiciIiUjIiR5I0YnIiInIyIlO0dGJyIiKS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzU=

coeff(coeff(Lser(n),x,1),y,0);

LC4qJiImSC8iIiIiKUkidEc2IiIjNUYlRiUqJiMiJTtHIiIkRiUpRiciIilGJUYlKiYjIiR5I0YtRiUpRiciIidGJUYlKiYjIiNLRi1GJSlGJyIiJUYlRiUqJiMiIiZGLUYlKUYnIiIjRiVGJSNGPkYtRiU=

series(subs(Zm=Zmser(n),B01Zm),t,n+1);

Ky9JInRHNiIjISIiIiIkIiIhRiUiIiNGJSIiJSMiI0JGJyIiJyMiJHElRiciIiktSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

coeff(coeff(Bser(n),x,0),y,1);

LC4qJiIlREMiIiIpSSJ0RzYiIiM1RiVGJSomIyIkcSUiIiRGJSlGJyIiKUYlRiUqJiMiI0JGLUYlKUYnIiInRiVGJSomI0YlRi1GJSlGJyIiJUYlISIiKiZGNkYlKUYnIiIjRiVGOUY2Rjk=

<Text-field style="Heading 2" layout="Heading 2">Degrees: 72 for L(x,y), B(x,y) and P(x,y)</Text-field>

eqL;

expression of E:=2K(x,y)L(x,y)

E1:=collect(subs(L(x,y)=0,eqL),[L,B,B01],factor);;

LCwqKiwoKigiIiMiIiJJInRHNiJGJylJInlHRilGJkYnRicqJkkieEdGKUYnRitGJyEiIiomRiZGJ0YoRidGJ0YnLUkiTEdGKTYkIiIhRitGJ0YtRi5GK0YuRi4qLkYmRidGKEYnLCYqJClGLUYmRidGJ0YnRidGJy1GMTYkRi1GM0YnRi1GLkYrRi5GLioqRihGJywmKiRGKkYnRidGJ0YnRictSSJCR0YpRjJGJ0YqRi5GJyooRihGJ0kkQjAxR0YpRidGK0YuRi4qJiMiIiUiIiRGJ0YtRidGJw==

We will determine the degree of E over Q(t) when x=2 and y=3

E2:=subs(x=2,y=3,E1);

LCwqKCMiIiIiIidGJSwmKiYiIz9GJUkidEc2IkYlRiVGJiEiIkYlLUkiTEdGKzYkIiIhIiIkRiVGLCooIyIiJkYxRiVGKkYlLUYuNiQiIiNGMEYlRiwqKCMiIzUiIipGJUYqRiUtSSJCR0YrRi9GJUYlKigjRiVGMUYlRipGJUkkQjAxR0YrRiVGLCMiIilGMUYl

We have an expression of B01 in terms of Zm, and then cubic equations for L(0,x) (resp. L(x,0), B(0,x)) over Q(x,T) (resp. Q(x,Z))

algL0xftxTm:=(4*(x^2+1))*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*L0xftx^3+(4*(x-1))*(x+1)*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*L0xftx^2+(x^2+1)*(Tm-1)*(Tm^3*x^4+9*Tm^2*x^4+2*Tm^3*x^2+27*Tm*x^4-46*Tm^2*x^2+27*x^4+Tm^3+6*Tm*x^2+9*Tm^2-90*x^2+27*Tm+27)*L0xftx+16*x^2*(x-1)*(x+1)*(Tm-1)^2;

algB0xsxZm := 27*x^2*(x^2+1)^4*(Zm-1)^2*(Zm+1)^2*(Zm^2+3)^9*B0xsx^3+1728*Zm^3*x^2*(x^2+1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^6*(4*Zm^3*x^2+Zm^2*x^2-4*Zm^3+Zm^2-x^2-1)*B0xsx^2+768*Zm^6*(Zm^2+3)^3*(Zm^8*x^8-12*Zm^8*x^6+8*Zm^6*x^8-26*Zm^8*x^4+608*Zm^6*x^6+18*Zm^4*x^8+384*Zm^5*x^6-12*Zm^8*x^2-848*Zm^6*x^4-216*Zm^4*x^6-384*Zm^3*x^6+Zm^8+608*Zm^6*x^2-468*Zm^4*x^4+192*Zm^2*x^6-27*x^8-384*Zm^5*x^2+8*Zm^6-216*Zm^4*x^2+384*Zm^2*x^4-60*x^6+384*Zm^3*x^2+18*Zm^4+192*Zm^2*x^2-66*x^4-60*x^2-27)*B0xsx+(16384*(Zm^10*x^6+4*Zm^9*x^6+3*Zm^10*x^4+14*Zm^8*x^6-60*Zm^9*x^4+36*Zm^7*x^6+3*Zm^10*x^2-70*Zm^8*x^4+70*Zm^6*x^6+60*Zm^9*x^2+228*Zm^7*x^4+108*Zm^5*x^6+Zm^10-70*Zm^8*x^2+210*Zm^6*x^4+144*Zm^4*x^6-4*Zm^9-228*Zm^7*x^2-852*Zm^5*x^4+108*Zm^3*x^6+14*Zm^8+210*Zm^6*x^2-752*Zm^4*x^4+81*Zm^2*x^6-36*Zm^7+852*Zm^5*x^2-84*Zm^3*x^4+70*Zm^6-752*Zm^4*x^2+499*Zm^2*x^4-54*x^6-108*Zm^5+84*Zm^3*x^2+144*Zm^4+499*Zm^2*x^2-146*x^4-108*Zm^3+81*Zm^2-146*x^2-54))*Zm^9;

somme:=op(1,E2)+op(3,E2);

LCYqKCMiIiIiIidGJSwmKiYiIz9GJUkidEc2IkYlRiVGJiEiIkYlLUkiTEdGKzYkIiIhIiIkRiVGLCooIyIjNSIiKkYlRipGJS1JIkJHRitGL0YlRiU=

res1:=factor(resultant(subs(L(0,3)=L0xftx/t/x,x=3,som-somme),subs(x=3,algL0xftxTm),L0xftx)): nops(res1);

IiIk

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

NiUiIiMiIyMpRiM=

NiU8IjwlSSNUbUc2Ikkkc29tR0YmSSJ0R0YmPCNGKA==

res2:=op(2,res1):

res3:=factor(resultant(subs(B(0,3)=B0xsx*3,Tm=Zm^2,res2),subs(x=3,algB0xsxZm),B0xsx)): nops(res3);

IiIk

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiUiIiIiJEUkIiUrQg==

NiU8IjwlSSNabUc2Ikkkc29tR0YmSSJ0R0YmRiQ=

n:=16: normal(series(subs(Zm=Zmser(n),som=somme,L(0,3)=subs(x=0,y=3,Lser(n)),B(0,3)=subs(x=0,y=3,Bser(n)) ,x=2,y=3,op(2,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

normal(series(subs(Zm=Zmser(n),som=somme,L(0,3)=subs(x=0,y=3,Lser(n)),B(0,3)=subs(x=0,y=3,Bser(n)) ,x=2,y=3,op(3,res3)),t,n+1));

KzFJInRHNiIiPCFHXC9hWmF2MCMzYShwUiIjNSE9Z1AqUW95UUAkeVBOWFNNIiM2Ij4/KiozZjcheUQmKnpgJ3ljXSMiIzchP2d4NlsrI1sjKmVlSSpRZGY+IiM4IkA/JlJkIVs6TTYiek5wWEFkNyIjOSFAP0I6R0FpbUJZSUlpOE5bKSIjOi1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

res4:=op(2,res3):

algsom:=res4:

E3:=subs(L(0,3)=0,B(0,3)=0,E2)+som;

LCojIiIpIiIkIiIiKigjIiImRiVGJkkidEc2IkYmLUkiTEdGKzYkIiIjIiIhRiYhIiIqKCNGJkYlRiZGKkYmSSRCMDFHRitGJkYxSSRzb21HRitGJg==

B01Zm:= -(64/3)*Zm^3*(Zm^3-3*Zm^2+2*Zm+2)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

indets(algsom);

PCVJI1ptRzYiSSRzb21HRiRJInRHRiQ=

algLx0sxZm := 27*x^2*(x^2+1)^4*(Zm-1)^2*(Zm+1)^2*(Zm^2+3)^9*Lx0sx^3-864*Zm^3*x^2*(x^2+1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^6*(Zm^4*x^2+4*Zm^3*x^2+Zm^4+4*Zm^2*x^2-4*Zm^3+4*Zm^2-x^2-1)*Lx0sx^2+3072*Zm^6*(Zm^2+3)^3*(Zm^8*x^8+3*Zm^8*x^6+8*Zm^6*x^8+24*Zm^7*x^6+4*Zm^8*x^4+56*Zm^6*x^6+18*Zm^4*x^8+96*Zm^5*x^6+3*Zm^8*x^2-32*Zm^6*x^4+30*Zm^4*x^6-24*Zm^7*x^2-24*Zm^3*x^6+Zm^8+56*Zm^6*x^2+24*Zm^4*x^4+48*Zm^2*x^6-27*x^8-96*Zm^5*x^2+8*Zm^6+30*Zm^4*x^2+96*Zm^2*x^4-105*x^6+24*Zm^3*x^2+18*Zm^4+48*Zm^2*x^2-156*x^4-105*x^2-27)*Lx0sx-32768*Zm^9*(Zm^10*x^6+4*Zm^9*x^6+17*Zm^8*x^6+36*Zm^7*x^6-4*Zm^8*x^4+94*Zm^6*x^6+108*Zm^5*x^6+Zm^10-4*Zm^8*x^2-48*Zm^6*x^4+198*Zm^4*x^6-4*Zm^9-96*Zm^5*x^4+108*Zm^3*x^6+17*Zm^8-48*Zm^6*x^2-104*Zm^4*x^4+81*Zm^2*x^6-36*Zm^7+96*Zm^5*x^2+96*Zm^3*x^4+94*Zm^6-104*Zm^4*x^2+304*Zm^2*x^4-135*x^6-108*Zm^5-96*Zm^3*x^2+198*Zm^4+304*Zm^2*x^2-404*x^4-108*Zm^3+81*Zm^2-404*x^2-135);

res5:=factor(resultant(E-subs(L(2,0)=Lx0sx*x,x=2,B01=B01Zm,E3),subs(x=2, algLx0sxZm),Lx0sx)): nops(res5);

IiIk

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiUiIiMiJDIlRiM=

NiU8IjwmSSJFRzYiSSNabUdGJkkkc29tR0YmSSJ0R0YmPCNGJw==

res6:=op(2,res5):

res7:=factor(resultant(res6,algsom,som)): nops(res7);

IiIm

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NiciIiIiIiNGJEYkIiUjKikp

Nic8IjwjSSNabUc2IkYkRiQ8JUkiRUdGJkYlSSJ0R0Ym

res8:=op(5,res7):

degree(res8,t);

IiM9

res9:=factor(resultant(res8,algZm,Zm)): nops(res9);

IiIk

seq(nops(op(i,res9)),i=1..nops(res9)); seq(indets(op(i,res9)),i=1..nops(res9));

NiUiIiIiIiMiJSE9Jw==

NiU8IjwjSSJ0RzYiPCRJIkVHRiZGJQ==

res10:=op(3,res9): degree(%,E);

IiNz

let us check

n:=20: series(subs(E=2*K*Lser(n),x=2,y=3,res10),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

And now with B(x,y)

eqB;

expression of E:=2K(x,y)B(x,y)

E1:=collect(subs(B(x,y)=0,eqB),[L,B,L10],factor);;

LCwqKkkidEc2IiIiIiwmKiQpSSJ4R0YlIiIjRiZGJkYmRiZGJi1JIkxHRiU2JEYqIiIhRiZGKSEiIkYmKi5GK0YmRiRGJiwmKiQpSSJ5R0YlRitGJkYmRiZGJkYmLUkiQkdGJTYkRi9GNUYmRipGMEY1RjBGMCoqLCgqKEYrRiZGJEYmRilGJkYmKiZGKkYmRjVGJkYwKiZGK0YmRiRGJkYmRiYtRjdGLkYmRipGMEY1RjBGMCooRiRGJkkkTDEwR0YlRiZGKkYwRjAqJiNGKyIiJEYmRjVGJkYw

We will determine the degree of E over Q(t) when x=2 and y=3

E2:=subs(x=2,y=3,E1);

LCwqKCMiIiYiIiUiIiJJInRHNiJGJy1JIkxHRik2JCIiIyIiIUYnRicqKCMiIzUiIiRGJ0YoRictSSJCR0YpNiRGLkYyRichIiIqKCNGJyIiJ0YnLCYqJkYxRidGKEYnRidGOUY2RictRjRGLEYnRjYqKCNGJ0YtRidGKEYnSSRMMTBHRilGJ0Y2Ri1GNg==

We have an expression of B01 in terms of Zm, and then cubic equations for L(0,x) (resp. L(x,0), B(0,x)) over Q(x,T) (resp. Q(x,Z))

algLx0sxZm;

algBx0ftxTm := (x^2+1)*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*Bx0ftx^3+(x^2+2)*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*Bx0ftx^2+(2*Tm^4*x^4+3*Tm^4*x^2+40*Tm^2*x^4-56*Tm^3*x^2-24*Tm*x^4+Tm^4+58*Tm^2*x^2-18*x^4+8*Tm^3-24*Tm*x^2+18*Tm^2-45*x^2-27)*Bx0ftx+(Tm-1)^2*(Tm^2+4*x^2-6*Tm+5)*x^2;

somme:=op(1,E2)+op(3,E2);

LCYqKCMiIiYiIiUiIiJJInRHNiJGJy1JIkxHRik2JCIiIyIiIUYnRicqKCNGJyIiJ0YnLCYqJiIjNUYnRihGJ0YnRjEhIiJGJy1JIkJHRilGLEYnRjU=

res1:=factor(resultant(subs(L(2,0)=Lx0sx*x,x=2,som-somme),subs(x=2,algLx0sxZm),Lx0sx)): nops(res1);

IiQhRw==

res2:=(res1):indets(%);

PCVJI1ptRzYiSSRzb21HRiRJInRHRiQ=

res3:=factor(resultant(subs(B(2,0)=Bx0ftx/t/x,x=2,res2),subs(x=2,Tm=Zm^2,algBx0ftxTm),Bx0ftx)): nops(res3);

IiIl

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiYiIiIiJFUkIiVSQiIiIw==

NiY8IjwlSSNabUc2Ikkkc29tR0YmSSJ0R0YmRiQ8I0Yo

n:=16: normal(series(subs(Zm=Zmser(n),som=somme,L(2,0)=subs(x=2,y=0,Lser(n)),B(2,0)=subs(x=2,y=0,Bser(n)) ,x=2,y=3,op(2,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

normal(series(subs(Zm=Zmser(n),som=somme,L(2,0)=subs(x=2,y=0,Lser(n)),B(2,0)=subs(x=2,y=0,Bser(n)) ,x=2,y=3,op(3,res3)),t,n+1));

KzNJInRHNiIhN2dQaHlmOVUlKj4sQCIiKiI4K2s/eicqPUs7KnpeSiIjNSE5IUdwKDMkPikqPVpOIiopPiIjNiI5P1JOZXo5VldlOSR6KiIjNyE6KytfQmwiUi0wRUJWRiIjOCI6Z2AmPUQ6JCpmIT12XjQmIiM5Ijs/emxLTSg9NWtpVUwiRyIjOi1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

res4:=op(2,res3):

algsom:=res4:

E3:=subs(L(2,0)=0,B(2,0)=0,E2)+som;

LCoiIiMhIiIqKCMiIzUiIiQiIiJJInRHNiJGKS1JIkJHRis2JCIiIUYoRilGJCooI0YpRiNGKUYqRilJJEwxMEdGK0YpRiRJJHNvbUdGK0Yp

L10Zm:=(32/3)*Zm^3*(Zm^3-3*Zm^2+5*Zm+5)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

algB0xsxZm;

res5:=factor(resultant(E-subs(B(0,3)=B0xsx*x,x=3,L10=L10Zm,E3),subs(x=3,algB0xsxZm),B0xsx)): nops(res5);

IiIk

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiUiIiIiJD8lIiIj

NiU8IjwmSSJFRzYiSSNabUdGJkkkc29tR0YmSSJ0R0YmPCNGJw==

res6:=op(2,res5):

res7:=factor(resultant(res6,algsom,som)): nops(res7);

IiIm

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NiciIiIiIiNGJEYkIiVCKik=

Nic8IjwjSSNabUc2IkYkRiQ8JUkiRUdGJkYlSSJ0R0Ym

res8:=op(5,res7):

degree(res8,t);

IiM9

res9:=factor(resultant(res8,algZm,Zm)): nops(res9);

IiIk

seq(nops(op(i,res9)),i=1..nops(res9)); seq(indets(op(i,res9)),i=1..nops(res9));

NiUiIiIiIiMiJXlo

NiU8IjwjSSJ0RzYiPCRJIkVHRiZGJQ==

res10:=op(3,res9): degree(%,E);

IiNz

let us check

n:=20: series(subs(E=2*K*Bser(n),x=2,y=3,res10),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

And finally P(x,y)

Pexpr;eqL; eqB;

LCoqJi1JIkxHNiI2JEkieEdGJkkieUdGJiIiIkYoISIiRioqJi1GJTYkIiIhRilGKkYoRitGKyomLUkiQkdGJkYnRipGKUYrRioqJi1GMjYkRihGL0YqRilGK0Yr

eqP:=collect(2*K*(-P(x,y)+Pexpr)+eqL/x+eqB/y,[L,B,P,L10,B01],factor);

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

expression of E:=2K(x,y)P(x,y)

E1:=collect(subs(P(x,y)=0,eqP),[P, L,B,L10,B01],factor);;

LDAqKkkieEc2IiEiIiwoKioiIiMiIiJJInRHRiVGKkYkRiopSSJ5R0YlRilGKkYqKihGKUYqRitGKkYkRipGKkYtRiZGKkYtRiYtSSJMR0YlNiQiIiFGLUYqRioqLEYrRiosJiokKUYkRilGKkYqRipGKkYqRjZGJkYtRiYtRjA2JEYkRjJGKkYmKixGK0YqLCYqJEYsRipGKkYqRipGKkYsRiZGJEYmLUkiQkdGJUYxRipGJioqRi1GJiwoKipGKUYqRitGKkY2RipGLUYqRioqKEYpRipGK0YqRi1GKkYqRiRGJkYqRiRGJi1GPUY4RipGKiNGKSIiJEYqKipGK0YqRiRGJkYtRiZJJEwxMEdGJUYqRiYqKkYrRipGJEYmRi1GJkkkQjAxR0YlRipGJg==

We will determine the degree of E over Q(t) when x=2 and y=3

E2:=subs(x=2,y=3,E1);

LDAqKCMiIiIiIidGJSwmKiYiI1NGJUkidEc2IkYlRiUiIiQhIiJGJS1JIkxHRis2JCIiIUYsRiVGJSooIyIiJiIjN0YlRipGJS1GLzYkIiIjRjFGJUYtKigjRjQiIipGJUYqRiUtSSJCR0YrRjBGJUYtKihGJEYlLCYqJiIjSUYlRipGJUYlRjhGLUYlLUY9RjdGJUYlI0Y4RixGJSooRiRGJUYqRiVJJEwxMEdGK0YlRi0qKEYkRiVGKkYlSSRCMDFHRitGJUYt

Two sums

algLx0sxZm;

somme2:=op(2,E2)+op(4,E2);

LCYqKCMiIiYiIzciIiJJInRHNiJGJy1JIkxHRik2JCIiIyIiIUYnISIiKigjRiciIidGJywmKiYiI0lGJ0YoRidGJ0YtRi9GJy1JIkJHRilGLEYnRic=

res1:=factor(resultant(subs(L(2,0)=Lx0sx*x,x=2,som2-somme2),subs(x=2,algLx0sxZm),Lx0sx)): nops(res1);

IiQhRw==

res2:=(res1):indets(%);

PCVJI1ptRzYiSSVzb20yR0YkSSJ0R0Yk

res3:=factor(resultant(subs(B(2,0)=Bx0ftx/t/x,x=2,res2),subs(x=2,Tm=Zm^2,algBx0ftxTm),Bx0ftx)): nops(res3);

IiIl

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiYiIiMiJFUkIiVSQkYj

NiY8IjwlSSNabUc2Ikklc29tMkdGJkkidEdGJkYkPCNGKA==

n:=16: normal(series(subs(Zm=Zmser(n),som2=somme2,L(2,0)=subs(x=2,y=0,Lser(n)),B(2,0)=subs(x=2,y=0,Bser(n)) ,x=2,y=3,op(2,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

normal(series(subs(Zm=Zmser(n),som2=somme2,L(2,0)=subs(x=2,y=0,Lser(n)),B(2,0)=subs(x=2,y=0,Bser(n)) ,x=2,y=3,op(3,res3)),t,n+1));

KzNJInRHNiIhOFNRXzJRSnpcejUqPSIiKiI6K0NZWm94PnBsUEFIIiIjNSE7P04vV3I2Zy9COlkrRiIjNiI8Pz5fdCpRIW9DVHlcImU+IiM3ITwrKy8kPVMkb3FkPEFYNSQiIzgiPGdgTEdGbSE0LHRIUiZcJiIjOSI9IUdAUkgmUmo9dVQlKj4xYiIjOi1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

res4:=op(2,res3):

algsom2:=res4:

The second sum

somme3:=op(1,E2)+op(3,E2);

LCYqKCMiIiIiIidGJSwmKiYiI1NGJUkidEc2IkYlRiUiIiQhIiJGJS1JIkxHRis2JCIiIUYsRiVGJSooIyIiJiIiKkYlRipGJS1JIkJHRitGMEYlRi0=

res1:=factor(resultant(subs(L(0,3)=L0xftx/t/x,x=3,som3-somme3),subs(x=3,algL0xftxTm),L0xftx)): nops(res1);

IiIk

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

NiUiIiMiIyMpRiM=

NiU8IjwlSSNUbUc2Ikklc29tM0dGJkkidEdGJjwjRig=

res2:=op(2,res1):

res3:=factor(resultant(subs(B(0,3)=B0xsx*3,Tm=Zm^2,res2),subs(x=3,algB0xsxZm),B0xsx)): nops(res3);

IiIk

seq(nops(op(i,res3)),i=1..nops(res3)); seq(indets(op(i,res3)),i=1..nops(res3));

NiUiIiIiJEUkIiUsQg==

NiU8IjwlSSNabUc2Ikklc29tM0dGJkkidEdGJkYk

n:=16: normal(series(subs(Zm=Zmser(n),som3=somme3,L(0,3)=subs(x=0,y=3,Lser(n)),B(0,3)=subs(x=0,y=3,Bser(n)) ,x=2,y=3,op(2,res3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM7

normal(series(subs(Zm=Zmser(n),som3=somme3,L(0,3)=subs(x=0,y=3,Lser(n)),B(0,3)=subs(x=0,y=3,Bser(n)) ,x=2,y=3,op(3,res3)),t,n+1));

KzFJInRHNiIiOz9CNiYpPScpUTkwX1FDKioiIzUhPWdQKlFveVFAJHlQTlhTTSIjNiI+IVtjZiZbWkkoZkwwNE9sVSIjNyE/Z3goeT4rUzJXV1BEREdHJCIjOCJAISkpKTR2eF1VPG9FJVJVbTdNIiM5IUE/QiJ6cD1fbl5QL1U1PmM7JCIjOi1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

res4:=op(2,res3):

algsom3:=res4:

E3:=subs(L(2,0)=0,B(2,0)=0,L(0,3)=0,B(0,3)=0,E2)+som2+som3;

LCwjIiIjIiIkIiIiKigjRiYiIidGJkkidEc2IkYmSSRMMTBHRitGJiEiIiooRihGJkYqRiZJJEIwMUdGK0YmRi1JJXNvbTJHRitGJkklc29tM0dGK0Ym

res5:=factor(resultant(subs(L10=L10Zm,B01=B01Zm,E-E3),algsom2,som2)): nops(res5);

IiIk

seq(nops(op(i,res5)),i=1..nops(res5)); seq(indets(op(i,res5)),i=1..nops(res5));

NiUiIiIiJCcpKSIiIw==

NiU8IjwmSSJFRzYiSSNabUdGJkklc29tM0dGJkkidEdGJjwjRic=

res6:=op(2,res5):

res7:=factor(resultant(res6,algsom3,som3)): nops(res7);

IiIn

seq(nops(op(i,res7)),i=1..nops(res7)); seq(indets(op(i,res7)),i=1..nops(res7));

NigiIiIiIiNGJEYkRiQiJVMhKg==

Nig8IjwjSSJ0RzYiPCNJI1ptR0YmRidGJzwlSSJFR0YmRihGJQ==

res8:=op(6,res7):

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

LCwqJClJI1ptRzYiIiIpIiIiISIiKigiJGMjRigpRiUiIidGKClJInRHRiYiIiNGKEYoKiZGJ0YoRixGKEYpKiYiIz1GKClGJSIiJUYoRikiI0ZGKA==

res9:=factor(resultant(res8,algZm,Zm)): nops(res9);

IiIk

seq(nops(op(i,res9)),i=1..nops(res9)); seq(indets(op(i,res9)),i=1..nops(res9));

NiUiIiIiIiMiJWUnKg==

NiU8IjwjSSJ0RzYiPCRJIkVHRiZGJQ==

res10:=op(3,res9): degree(%,E);

IiNz

let us check

n:=20: series(subs(E=2*K*Pser(n),x=2,y=3,res10),t,n+1);

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiNA

<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(x,0) is a rational function in Z

algLx0sxZm:=27*x^2*(x^2+1)^4*(Zm-1)^2*(Zm+1)^2*(Zm^2+3)^9*Lx0sx^3-864*Zm^3*x^2*(x^2+1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^6*(Zm^4*x^2+4*Zm^3*x^2+Zm^4+4*Zm^2*x^2-4*Zm^3+4*Zm^2-x^2-1)*Lx0sx^2+3072*Zm^6*(Zm^2+3)^3*(Zm^8*x^8+3*Zm^8*x^6+8*Zm^6*x^8+24*Zm^7*x^6+4*Zm^8*x^4+56*Zm^6*x^6+18*Zm^4*x^8+96*Zm^5*x^6+3*Zm^8*x^2-32*Zm^6*x^4+30*Zm^4*x^6-24*Zm^7*x^2-24*Zm^3*x^6+Zm^8+56*Zm^6*x^2+24*Zm^4*x^4+48*Zm^2*x^6-27*x^8-96*Zm^5*x^2+8*Zm^6+30*Zm^4*x^2+96*Zm^2*x^4-105*x^6+24*Zm^3*x^2+18*Zm^4+48*Zm^2*x^2-156*x^4-105*x^2-27)*Lx0sx-(32768*(Zm^10*x^6+4*Zm^9*x^6+17*Zm^8*x^6+36*Zm^7*x^6-4*Zm^8*x^4+94*Zm^6*x^6+108*Zm^5*x^6+Zm^10-4*Zm^8*x^2-48*Zm^6*x^4+198*Zm^4*x^6-4*Zm^9-96*Zm^5*x^4+108*Zm^3*x^6+17*Zm^8-48*Zm^6*x^2-104*Zm^4*x^4+81*Zm^2*x^6-36*Zm^7+96*Zm^5*x^2+96*Zm^3*x^4+94*Zm^6-104*Zm^4*x^2+304*Zm^2*x^4-135*x^6-108*Zm^5-96*Zm^3*x^2+198*Zm^4+304*Zm^2*x^2-404*x^4-108*Zm^3+81*Zm^2-404*x^2-135))*Zm^9;

LCoqMCIjRiIiIilJInhHNiIiIiNGJSksJiokRiZGJUYlRiVGJSIiJUYlKSwmSSNabUdGKEYlRiUhIiJGKUYlKSwmRjBGJUYlRiVGKUYlKSwmKiQpRjBGKUYlRiUiIiRGJSIiKkYlKUkmTHgwc3hHRihGOEYlRiUqNCIkaylGJSlGMEY4RiVGJkYlKUYrRilGJUYvRiVGM0YlKUY1IiInRiUsMiomKUYwRi1GJUYmRiVGJSooRi1GJUY+RiVGJkYlRiUqJEZERiVGJSooRi1GJUY3RiVGJkYlRiUqJkYtRiVGPkYlRjEqJkYtRiVGN0YlRiVGLEYxRiVGMUYlKUY7RilGJUYxKiwiJXNJRiUpRjBGQUYlKUY1RjhGJSxmbiomKUYwIiIpRiUpRidGUkYlRiUqKEY4RiVGUUYlKUYnRkFGJUYlKihGUkYlRk1GJUZTRiVGJSooIiNDRiUpRjAiIihGJUZVRiVGJSooRi1GJUZRRiUpRidGLUYlRiUqKCIjY0YlRk1GJUZVRiVGJSooIiM9RiVGREYlRlNGJUYlKigiIycqRiUpRjAiIiZGJUZVRiVGJSooRjhGJUZRRiVGJkYlRiUqKCIjS0YlRk1GJUZmbkYlRjEqKCIjSUYlRkRGJUZVRiVGJSooRlhGJUZZRiVGJkYlRjEqKEZYRiVGPkYlRlVGJUYxKiRGUUYlRiUqKEZobkYlRk1GJUYmRiVGJSooRlhGJUZERiVGZm5GJUYlKigiI1tGJUY3RiVGVUYlRiUqJkYkRiVGU0YlRjEqKEZcb0YlRl1vRiVGJkYlRjEqJkZSRiVGTUYlRiUqKEZjb0YlRkRGJUYmRiVGJSooRlxvRiVGN0YlRmZuRiVGJSomIiQwIkYlRlVGJUYxKihGWEYlRj5GJUYmRiVGJSomRmpuRiVGREYlRiUqKEZqb0YlRjdGJUYmRiVGJSomIiRjIkYlRmZuRiVGMSomRmFwRiVGJkYlRjFGJEYxRiVGO0YlRiUqKCImb0YkRiUsYG8qJilGMCIjNUYlRlVGJUYlKihGLUYlKUYwRjlGJUZVRiVGJSooIiM8RiVGUUYlRlVGJUYlKigiI09GJUZZRiVGVUYlRiVGZW5GMSooIiMlKkYlRk1GJUZVRiVGJSooIiQzIkYlRl1vRiVGVUYlRiUqJEZccUYlRiUqKEYtRiVGUUYlRiZGJUYxKihGam9GJUZNRiVGZm5GJUYxKigiJCk+RiVGREYlRlVGJUYlKiZGLUYlRl9xRiVGMSooRlxvRiVGXW9GJUZmbkYlRjEqKEZncUYlRj5GJUZVRiVGJSomRmFxRiVGUUYlRiUqKEZqb0YlRk1GJUYmRiVGMSooIiQvIkYlRkRGJUZmbkYlRjEqKCIjIilGJUY3RiVGVUYlRiUqJkZjcUYlRllGJUYxRlxwRiUqKEZcb0YlRj5GJUZmbkYlRiUqJkZlcUYlRk1GJUYlKihGY3JGJUZERiVGJkYlRjEqKCIkLyRGJUY3RiVGZm5GJUYlKiYiJE4iRiVGVUYlRjEqJkZncUYlRl1vRiVGMSooRlxvRiVGPkYlRiZGJUYxKiZGXHJGJUZERiVGJSooRltzRiVGN0YlRiZGJUYlKiYiJC8lRiVGZm5GJUYxKiZGZ3FGJUY+RiVGMSomRmVyRiVGN0YlRiUqJkZjc0YlRiZGJUYxRl1zRjFGJUZfcUYlRjE=

factor(subs(x=0,algLx0sxZm));degree(%,Lx0sx);

LCQqLCIlQzUiIiIpSSNabUc2IiIiJ0YlLCZGJ0YlRiUhIiJGJSksJiokKUYnIiIjRiVGJSIiJEYlRjFGJSw6KihGMUYlSSZMeDBzeEdGKEYlKUYnIiIoRiVGJSooRjFGJUY0RiVGJkYlRiUqKCIjRkYlRjRGJSlGJyIiJkYlRiUqJiIjS0YlRiZGJUYrKihGOUYlRjRGJSlGJyIiJUYlRiUqJiIjJypGJUY6RiVGJSooIiMiKUYlRjRGJSlGJ0YxRiVGJSomIiRnIkYlRj9GJUYrKihGREYlRjRGJUYvRiVGJSomRkdGJUZFRiVGKyooRkRGJUY0RiVGJ0YlRiUqJkZERiVGNEYlRiVGJUYl

IiIi

factor([solve(%%,Lx0sx)]);

NyMsJCosIyIjSyIiJCIiIilJI1ptRzYiIiIkIiIiLCoqJClJI1ptRzYiIiIkIiIiIiIiKiYiIiQiIiIpSSNabUc2IiIiIyIiIiEiIiomIiImIiIiSSNabUc2IiIiIiIiIiIiJiIiIiIiIiwmSSNabUc2IiIiIiIiIiIiIiEiIiksJiokKUkjWm1HNiIiIiMiIiIiIiIiIiQiIiIiIiQhIiIiIiI=

L10Zm:=op(%);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

Now for txL(0,x) : coefficients in Q(T)

algL0xftxTm;

map(factor,op(2,algeqtoseries(algL0xftxTm,x,L0xftx,4,true)));

KylJInhHNiIsJCooIiM7IiIiLCZJI1RtR0YkRihGKCEiIkYoKSwmRipGKCIiJEYoRi5GK0YoIiIjLCQqLCIja0YoKUYpRi9GKCwmRipGKCIiKEYrRigsJkYqRihGKEYoRigpRi0iIidGK0YrIiIlLUkiT0clKnByb3RlY3RlZEc2I0YoRjg=

factor(subs(x=0,algL0xftxTm)); degree(%,L0xftx);

KipJJ0wweGZ0eEc2IiIiIiwmSSNUbUdGJEYlRiUhIiJGJSksJkYnRiUiIiRGJUYrRiUpLCYqJiIiI0YlRiNGJUYlRiVGKEYvRiU=

IiIk

Now for B(0,x)/x

algB0xsxZm;

factor(subs(x=0,algB0xsxZm));degree(%,B0xsx);

LCQqLCIkYyMiIiIpSSNabUc2IiIiJ0YlLCZGJ0YlRiUhIiJGJSksJiokKUYnIiIjRiVGJSIiJEYlRjFGJSw6KihGMUYlSSZCMHhzeEdGKEYlKUYnIiIoRiVGJSooRjFGJUY0RiVGJkYlRiUqKCIjRkYlRjRGJSlGJyIiJkYlRiUqJiIja0YlRiZGJUYlKihGOUYlRjRGJSlGJyIiJUYlRiUqJiIkIz5GJUY6RiVGKyooIiMiKUYlRjRGJSlGJ0YxRiVGJSomIiRHIkYlRj9GJUYlKihGREYlRjRGJUYvRiVGJSomRkdGJUZFRiVGJSooRkRGJUY0RiVGJ0YlRiUqJkZERiVGNEYlRiVGJUYl

IiIi

factor([solve(%%,B0xsx)]);

NyMsJCosIyIjayIiJCIiIilJI1ptRzYiIiIkIiIiLCoqJClJI1ptRzYiIiIkIiIiIiIiKiYiIiQiIiIpSSNabUc2IiIiIyIiIiEiIiomIiIjIiIiSSNabUc2IiIiIiIiIiIiIyIiIiIiIiwmSSNabUc2IiIiIiIiIiIiIiEiIiksJiokKUkjWm1HNiIiIiMiIiIiIiIiIiQiIiIiIiQhIiIhIiI=

Now for txB(x,0) : coefficients in Q(T) again

algBx0ftxTm;

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

factor(subs(x=0,algBx0ftxTm));degree(%,Bx0ftx);

KipJJ0J4MGZ0eEc2IiIiIiwmSSNUbUdGJEYlRiUhIiJGJSksJkYnRiUiIiRGJUYrRiUpLCZGI0YlRiVGJSIiI0Yl

IiIk

map(factor,op(2,algeqtoseries(algBx0ftxTm,x,Bx0ftx,5,true)));

KytJInhHNiIsJCooLCZJI1RtR0YkIiIiIiImISIiRikpLCZGKEYpRilGKyIiI0YpKSwmRihGKSIiJEYpRjFGK0YrRi4qKCwqKiQpRihGMUYpRikqJiIjVkYpKUYoRi5GKUYrKiYiJEoiRilGKEYpRikiJG4iRilGKSlGLUYxRikpRjAiIidGKyIiJSwkKigsLiokKUYoRipGKUYpKiYiJDgiRikpRihGP0YpRisqJiIlUTxGKUY1RilGKSomIiUhcCNGKUY4RilGKyomIiVyJCpGKUYoRilGKyIlRFxGK0YpKUYtRj9GKSlGMCIiKkYrRitGPi1JIk9HJSpwcm90ZWN0ZWRHNiNGKSIiKA==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-1,1) and (-1,3) and (-1,5): the series L01, L03 and L05 (in Q(T))</Text-field>

algeqtoseries(algL0xftxTm,x,L0xftx,2,true);

NyQsKCMiIiIiIiNGJSomLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsLiomLCoqJClJI1RtR0YtIiIlRiVGJSomIiIpRiUpRjQiIiRGJUYlKiYiIz1GJSlGNEYmRiVGJSIjRiEiIkYlKUkjX1pHRipGJkYlRiVGMkYlRjZGJSomIiM5RiVGPEYlRiUqJkY3RiVGNEYlRj5GJUYlRiVJInhHRi1GJUYlLUkiT0dGKzYjKiQpRkQjRjlGJkYlRiUrJ0ZELCQqKCIjO0YlLCZGNEYlRiVGPkYlLCoqJEY4RiVGJSomIiIqRiVGPEYlRiUqJkY9RiVGNEYlRiVGPUYlRj5GJUYmLUZGNiNGJUY1

with(gfun): ser:=map(factor,op(2,algeqtoseries(algL0xftxTm,x,L0xftx,5,true)));

KytJInhHNiIsJCooIiM7IiIiLCZJI1RtR0YkRihGKCEiIkYoKSwmRipGKCIiJEYoRi5GK0YoIiIjLCQqLCIja0YoKUYpRi9GKCwmRipGKCIiKEYrRigsJkYqRihGKEYoRigpRi0iIidGK0YrIiIlLCQqKkYnRigpRilGLkYoLCwqJiIiKkYoKUYqRjlGKEYoKiYiJC8jRigpRipGLkYoRisqJiIkNSZGKClGKkYvRihGKComIiV3OUYoRipGKEYoIiRwKEYoRigpRi1GP0YrRihGOC1JIk9HJSpwcm90ZWN0ZWRHNiNGKEY1

L01ftTm:=coeff(ser,x,2); L03ftTm:=factor(coeff(ser,x,4)); L05ftTm:=factor(coeff(ser,x,6));

LCQqKCIjOyIiIiwmSSNUbUc2IkYlRiUhIiJGJSksJkYnRiUiIiRGJUYsRilGJQ==

LCQqLCIjayIiIiksJkkjVG1HNiJGJUYlISIiIiIjRiUsJkYoRiUiIihGKkYlLCZGKEYlRiVGJUYlKSwmRihGJSIiJEYlIiInRipGKg==

LCQqKiIjOyIiIiksJkkjVG1HNiJGJUYlISIiIiIkRiUsLComIiIqRiUpRigiIiVGJUYlKiYiJC8jRiUpRihGK0YlRioqJiIkNSZGJSlGKCIiI0YlRiUqJiIldzlGJUYoRiVGJSIkcChGJUYlKSwmRihGJUYrRiVGLkYqRiU=

coeff(coeff(Lser(n),x,0),y,1);

LDIqJiIpd1xhNSIiIilJInRHNiIiIzpGJUYlKiYiJ15kIylGJSlGJyIjOEYlRiUqJiImd28nRiUpRiciIzZGJUYlKiYiJW5jRiUpRiciIipGJUYlKiYiJDcmRiUpRiciIihGJUYlKiYiI15GJSlGJyIiJkYlRiUqJiIiJ0YlKUYnIiIkRiVGJUYnRiU=

n:=16:normal(series(subs(Tm=Tmser(n),L01ftTm),t,n+1));

KzNJInRHNiIiIiIiIiMiIiciIiUiI15GJyIkNyYiIikiJW5jIiM1IiZ3byciIzciJ15kIykiIzktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzs=

Cm11ftsol:=L01ftTm; Cm13ftsol:=L03ftTm; Cm15ftsol:=L05ftTm;

LCQqKCIjOyIiIiwmSSNUbUc2IkYlRiUhIiJGJSksJkYnRiUiIiRGJUYsRilGJQ==

LCQqLCIjayIiIiksJkkjVG1HNiJGJUYlISIiIiIjRiUsJkYoRiUiIihGKkYlLCZGKEYlRiVGJUYlKSwmRihGJSIiJEYlIiInRipGKg==

LCQqKiIjOyIiIiksJkkjVG1HNiJGJUYlISIiIiIkRiUsLComIiIqRiUpRigiIiVGJUYlKiYiJC8jRiUpRihGK0YlRioqJiIkNSZGJSlGKCIiI0YlRiUqJiIldzlGJUYoRiVGJSIkcChGJUYlKSwmRihGJUYrRiVGLkYqRiU=

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

LDIqJiIpd1xhNSIiIilJInRHNiIiIzpGJUYlKiYiJ15kIylGJSlGJyIjOEYlRiUqJiImd28nRiUpRiciIzZGJUYlKiYiJW5jRiUpRiciIipGJUYlKiYiJDcmRiUpRiciIihGJUYlKiYiI15GJSlGJyIiJkYlRiUqJiIiJ0YlKUYnIiIkRiVGJUYnRiU=

n:=16:normal(series(subs(Tm=Tmser(n),Cm11ftsol),t,n+1));

KzNJInRHNiIiIiIiIiMiIiciIiUiI15GJyIkNyYiIikiJW5jIiM1IiZ3byciIzciJ15kIykiIzktSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzs=

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

LDAqJiIpT2JuOCIiIilJInRHNiIiIzpGJUYlKiYiKCEpeS0iRiUpRiciIzhGJUYlKiYiJmQoeUYlKUYnIiM2RiVGJSomIiVvaEYlKUYnIiIqRiVGJSomIiQlXEYlKUYnIiIoRiVGJSomIiNTRiUpRiciIiZGJUYlKiYiIiRGJSlGJ0Y/RiVGJQ==

n:=16:normal(series(subs(Tm=Tmser(n),Cm13ftsol),t,n+1));

KzNJInRHNiIiIiQiIiUiI1MiIiciJCVcIiIpIiVvaCIjNSImZCh5IiM3IighKXktIiIjOSIpT2JuOCIjOy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIz0=

coeff(coeff(convert(Cser(n),polynom),x,-1),y,5);

LC4qJiIpcSlSMyIiIiIpSSJ0RzYiIiM6RiVGJSomIidaL3ZGJSlGJyIjOEYlRiUqJiImIVJeRiUpRiciIzZGJUYlKiYiJTxNRiUpRiciIipGJUYlKiYiJDUjRiUpRiciIihGJUYlKiYiIzVGJSlGJyIiJkYlRiU=

n:=16:normal(series(subs(Tm=Tmser(n),Cm15ftsol),t,n+1));

KzFJInRHNiIiIzUiIiciJDUjIiIpIiU8TUYlIiYhUl4iIzciJ1ovdiIjOSIpcSlSMyIiIzstSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM9

L01ftTm := (16*(Tm-1))/(Tm+3)^3;

LCQqKCIjOyIiIiwmSSNUbUc2IkYlRiUhIiJGJSksJkYnRiUiIiRGJUYsRilGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,0), (-4,0), (-6,0) (-8,0): the series L10, L30, L50 and L70</Text-field>

algeqtoseries(algLx0sxZm,x,Lx0sx,2,true);

NyMrJ0kieEc2IiwkKiojIiNLIiIkIiIiKUkjWm1HNiIiIiQiIiIsKiokKUkjWm1HNiIiIiQiIiIiIiIqJiIiJCIiIilJI1ptRzYiIiIjIiIiISIiKiYiIiYiIiJJI1ptRzYiIiIiIiIiIiImIiIiIiIiLDIqJClJI1ptRzYiIiIoIiIiIiIiKiQpSSNabUc2IiIiJyIiIiIiIiomIiIqIiIiKUkjWm1HNiIiIiYiIiIiIiIqJiIiKiIiIilJI1ptRzYiIiIlIiIiIiIiKiYiI0YiIiIpSSNabUc2IiIiJCIiIiIiIiomIiNGIiIiKUkjWm1HNiIiIiMiIiIiIiIqJiIjRiIiIkkjWm1HNiIiIiIiIiIiI0YiIiIhIiIiIiIiIiEtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIj

ser:=map(factor,op(1,algeqtoseries(algLx0sxZm,x,Lx0sx,7,true)));

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

L10Zm:=coeff(ser,x,0); L30Zm:=coeff(ser,x,2); L50Zm:=coeff(ser,x,4); L70Zm:=coeff(ser,x,6);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNHRicpRikiIidGJ0YnKiYiI1VGJylGKSIiJkYnRiwqJiIjeUYnKUYpIiIlRidGJyomIiRFIkYnRihGJ0YsKiZGNkYnKUYpIiIjRidGJyomIiRVIkYnRilGJ0YsIiNyRixGJywmRilGJ0YnRidGLCksJiokRkRGJ0YnRiZGJ0Y4RixGLA==

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsPiokKUYpIiM4RidGJyomIiM8RicpRikiIzdGJyEiIiomIiNvRicpRikiIzZGJ0YnKiYiJC8iRicpRikiIzVGJ0YzKiYiJCJRRicpRikiIipGJ0YnKiYiJXg2RicpRikiIilGJ0YzKiYiJSNSIkYnKUYpIiIoRidGJyomIiU3RkYnKUYpIiInRidGMyomIiUiXCVGJylGKSIiJkYnRicqJiIlKkgiRicpRikiIiVGJ0YzKiYiJSt2RidGKEYnRicqJiIlRzxGJylGKSIiI0YnRicqJiIlKmYlRidGKUYnRiciJUw6RidGJyksJkYpRidGJ0YzRllGJywmRilGJ0YnRidGMyksJiokRlhGJ0YnRiZGJ0Y/RjNGJw==

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

Cm20sol:=L10Zm+Q00/3:

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

LC4qJiImPDUiIiIiKUkidEc2IiIjNUYlRiUqJiIlLzVGJSlGJyIiKUYlRiUqJiIkLCJGJSlGJyIiJ0YlRiUqJiIjN0YlKUYnIiIlRiVGJSomIiIjRiUpRidGN0YlRiVGJUYl

n:=16: normal(series(subs(Zm=Zmser(n),Q00=subs(x=0,y=0,Qser(n)),Cm20sol),t,n+1));

KzVJInRHNiIiIiIiIiEiIiNGJyIjNyIiJSIkLCIiIiciJS81IiIpIiY8NSIiIzUiJ18hSCJGKCIodUxlIiIjOS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjOw==

Cm40sol:=L30Zm+Q20/3:

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

LDIqJiIpQzkibyIiIiIpSSJ0RzYiIiM7RiVGJSomIihnVEoiRiUpRiciIzlGJUYlKiYiJz1mNUYlKUYnIiM3RiVGJSomIiUkKSkpRiUpRiciIzVGJUYlKiYiJCd5RiUpRiciIilGJUYlKiYiI3ZGJSlGJyIiJ0YlRiUqJkY5RiUpRiciIiVGJUYlKiQpRiciIiNGJUYl

n:=16: normal(series(subs(Zm=Zmser(n),Q20=coeff(subs(y=0,convert(Qser(n),polynom)),x,2),Cm40sol),t,n+1));

KzNJInRHNiIiIiIiIiMiIikiIiUiI3YiIiciJCd5RiciJSQpKSkiIzUiJz1mNSIjNyIoZ1RKIiIjOS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjOw==

Cm60sol:=L50Zm+Q40/3:

coeff(coeff(convert(Cser(n),polynom),x,-6),y,0);

LDAqJiIpKSopZToiIiIiKUkidEc2IiIjO0YlRiUqJiInKSlRJylGJSlGJyIjOUYlRiUqJiImT2InRiUpRiciIzdGJUYlKiYiJVhdRiUpRiciIzVGJUYlKiYiJCNSRiUpRiciIilGJUYlKiYiI0lGJSlGJyIiJ0YlRiUqJiIiI0YlKUYnIiIlRiVGJQ==

n:=16: normal(series(subs(Zm=Zmser(n),Q40=coeff(subs(y=0,convert(Qser(n),polynom)),x,4),Cm60sol),t,n+1));

KzNJInRHNiIiIiMiIiUiI0kiIiciJCNSIiIpIiVYXSIjNSImT2InIiM3IicpKVEnKSIjOSIpKSopZToiIiM7LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjPQ==

Cm80sol:=L70Zm+Q60/3:

n:=16:coeff(coeff(convert(Cser(n),polynom),x,-8),y,0);

LC4qJiIoI3BsaSIiIilJInRHNiIiIztGJUYlKiYiJygzSSVGJSlGJyIjOUYlRiUqJiImYSFIRiUpRiciIzdGJUYlKiYiJSEqPUYlKUYnIiM1RiVGJSomIiQ3IkYlKUYnIiIpRiVGJSomIiImRiUpRiciIidGJUYl

n:=16: normal(series(subs(Zm=Zmser(n),Q60=coeff(subs(y=0,convert(Qser(n),polynom)),x,6),Cm80sol),t,n+1));

KzFJInRHNiIiIiYiIiciJDciIiIpIiUhKj0iIzUiJmEhSCIjNyInKDNJJSIjOSIoI3BsaSIjOy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIz0=

L10Zm := (32/3)*Zm^3*(Zm^3-3*Zm^2+5*Zm+5)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

L30Zm := -(32/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+28*Zm^6-42*Zm^5+78*Zm^4-126*Zm^3+28*Zm^2-142*Zm-71)/((Zm+1)*(Zm^2+3)^6);

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-2) and (0,-4) and (0,-6) and (0,-8): the series B01, B03 , B05, B07</Text-field>

algB0xsxZm:=27*x^2*(x^2+1)^4*(Zm-1)^2*(Zm+1)^2*(Zm^2+3)^9*B0xsx^3+1728*Zm^3*x^2*(x^2+1)^2*(Zm-1)*(Zm+1)*(Zm^2+3)^6*(4*Zm^3*x^2+Zm^2*x^2-4*Zm^3+Zm^2-x^2-1)*B0xsx^2+768*Zm^6*(Zm^2+3)^3*(Zm^8*x^8-12*Zm^8*x^6+8*Zm^6*x^8-26*Zm^8*x^4+608*Zm^6*x^6+18*Zm^4*x^8+384*Zm^5*x^6-12*Zm^8*x^2-848*Zm^6*x^4-216*Zm^4*x^6-384*Zm^3*x^6+Zm^8+608*Zm^6*x^2-468*Zm^4*x^4+192*Zm^2*x^6-27*x^8-384*Zm^5*x^2+8*Zm^6-216*Zm^4*x^2+384*Zm^2*x^4-60*x^6+384*Zm^3*x^2+18*Zm^4+192*Zm^2*x^2-66*x^4-60*x^2-27)*B0xsx+(16384*(Zm^10*x^6+4*Zm^9*x^6+3*Zm^10*x^4+14*Zm^8*x^6-60*Zm^9*x^4+36*Zm^7*x^6+3*Zm^10*x^2-70*Zm^8*x^4+70*Zm^6*x^6+60*Zm^9*x^2+228*Zm^7*x^4+108*Zm^5*x^6+Zm^10-70*Zm^8*x^2+210*Zm^6*x^4+144*Zm^4*x^6-4*Zm^9-228*Zm^7*x^2-852*Zm^5*x^4+108*Zm^3*x^6+14*Zm^8+210*Zm^6*x^2-752*Zm^4*x^4+81*Zm^2*x^6-36*Zm^7+852*Zm^5*x^2-84*Zm^3*x^4+70*Zm^6-752*Zm^4*x^2+499*Zm^2*x^4-54*x^6-108*Zm^5+84*Zm^3*x^2+144*Zm^4+499*Zm^2*x^2-146*x^4-108*Zm^3+81*Zm^2-146*x^2-54))*Zm^9;

with(gfun):

ser:=map(factor,op(algeqtoseries(algB0xsxZm,x,B0xsx,7,true)));

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

B01Zm:=coeff(ser,x,0); B03Zm:=coeff(ser,x,2); B05Zm:=coeff(ser,x,4); B07Zm:=coeff(ser,x,6);

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

LCQqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNERicpRikiIidGJ0YnKiYiI1tGJylGKSIiJkYnRiwqJiIjJCpGJylGKSIiJUYnRicqJiIjISpGJ0YoRidGLComIiN6RicpRikiIiNGJ0YnKiYiI3dGJ0YpRidGLCIjUUYsRicsJkYpRidGJ0YnRiwpLCYqJEZFRidGJ0YmRidGOEYsRic=

LCQqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsPiokKUYpIiM4RidGJyomIiM8RicpRikiIzdGJyEiIiomIiNsRicpRikiIzZGJ0YnKiYiJDgiRicpRikiIzVGJ0YzKiYiJGMlRicpRikiIipGJ0YnKiYiJEcqRicpRikiIilGJ0YzKiYiJWU5RicpRikiIihGJ0YnKiYiJSc9JEYnKUYpIiInRidGMyomIiUwSkYnKUYpIiImRidGJyomIiVwUkYnKUYpIiIlRidGMyomIiU4V0YnRihGJ0YnKiYiJDQqRicpRikiIiNGJ0YzKiYiJSF6I0YnRilGJ0YnIiRJKkYnRicpLCZGKUYnRidGM0ZZRicsJkYpRidGJ0YnRjMpLCYqJEZYRidGJ0YmRidGP0YzRjM=

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

C0m2sol:=coeff(ser,x,0)+1/3*Q00; C0m4sol:=coeff(ser,x,2)+1/3*Q02; C0m6sol:=coeff(ser,x,4)+1/3*Q04; C0m8sol:=coeff(ser,x,6)+1/3*Q06;

LCYqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjAqJiNGJ0YmRidJJFEwMEdGKkYnRic=

LCYqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNERicpRikiIidGJ0YnKiYiI1tGJylGKSIiJkYnRiwqJiIjJCpGJylGKSIiJUYnRicqJiIjISpGJ0YoRidGLComIiN6RicpRikiIiNGJ0YnKiYiI3dGJ0YpRidGLCIjUUYsRicsJkYpRidGJ0YnRiwpLCYqJEZFRidGJ0YmRidGOEYsRicqJiNGJ0YmRidJJFEwMkdGKkYnRic=

LCYqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsPiokKUYpIiM4RidGJyomIiM8RicpRikiIzdGJyEiIiomIiNsRicpRikiIzZGJ0YnKiYiJDgiRicpRikiIzVGJ0YzKiYiJGMlRicpRikiIipGJ0YnKiYiJEcqRicpRikiIilGJ0YzKiYiJWU5RicpRikiIihGJ0YnKiYiJSc9JEYnKUYpIiInRidGMyomIiUwSkYnKUYpIiImRidGJyomIiVwUkYnKUYpIiIlRidGMyomIiU4V0YnRihGJ0YnKiYiJDQqRicpRikiIiNGJ0YzKiYiJSF6I0YnRilGJ0YnIiRJKkYnRicpLCZGKUYnRidGM0ZZRicsJkYpRidGJ0YnRjMpLCYqJEZYRidGJ0YmRidGP0YzRjMqJiNGJ0YmRidJJFEwNEdGKkYnRic=

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

n:=12: normal(series(subs(Zm=Zmser(n),Q00=coeff(coeff(convert(Qser(n),polynom),x,0),y,0),C0m2sol),t,n+1));

Ky1JInRHNiIiIiIiIiUiIzsiIiciJEEjIiIpIiU4SSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

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

LCwqJiImMzUlIiIiKUkidEc2IiIjN0YlRiUqJiIlOElGJSlGJyIjNUYlRiUqJiIkQSNGJSlGJyIiKUYlRiUqJiIjO0YlKUYnIiInRiVGJSokKUYnIiIlRiVGJQ==

n:=14: normal(series(subs(Zm=Zmser(n),Q02=coeff(coeff(convert(Qser(n),polynom),x,0),y,2),C0m4sol),t,n+1));

Ky1JInRHNiIiIiMiIiciI1oiIikiJEopIiM1IiZUTCIiIzctSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

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

LCwqJiInKVwwIyIiIilJInRHNiIiIzlGJUYlKiYiJlRMIkYlKUYnIiM3RiVGJSomIiRKKUYlKUYnIiM1RiVGJSomIiNaRiUpRiciIilGJUYlKiYiIiNGJSlGJyIiJ0YlRiU=

n:=16: normal(series(subs(Zm=Zmser(n),Q04=coeff(coeff(convert(Qser(n),polynom),x,0),y,4),C0m6sol),t,n+1));

Ky9JInRHNiIiIiYiIikiJGMiIiM1IiVqTCIjNyImV0QnIiM5IihlJXo1IiM7LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjPQ==

coeff(coeff(convert(Cser(n),polynom),x,0),y,-6);

LCwqJiIoZSV6NSIiIilJInRHNiIiIztGJUYlKiYiJldEJ0YlKUYnIiM5RiVGJSomIiVqTEYlKUYnIiM3RiVGJSomIiRjIkYlKUYnIiM1RiVGJSomIiImRiUpRiciIilGJUYl

n:=16: normal(series(subs(Zm=Zmser(n),Q06=coeff(coeff(convert(Qser(n),polynom),x,0),y,6),C0m8sol),t,n+1));

Ky1JInRHNiIiIzkiIzUiJFkmIiM3IiYzUiJGJSInSFhIIiM7LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjPQ==

coeff(coeff(convert(Cser(n),polynom),x,0),y,-8);

LCoqJiInSFhIIiIiKUkidEc2IiIjO0YlRiUqJiImM1IiRiUpRiciIzlGJUYlKiYiJFkmRiUpRiciIzdGJUYlKiZGLUYlKUYnIiM1RiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-2,2): the series L12 </Text-field>

eqL;

LC4qKiMiIiIiIiNGJSwoKihGJkYlSSJ0RzYiRiUpSSJ5R0YqRiZGJUYlKiZJInhHRipGJUYsRiUhIiIqJkYmRiVGKUYlRiVGJS1JIkxHRio2JCIiIUYsRiVGLEYvRi8qKkYpRiUsJiokKUYuRiZGJUYlRiVGJUYlLUYyNiRGLkY0RiVGLEYvRi8qKCwsKihGKUYlRjhGJUYrRiVGJSomRilGJUY4RiVGJSomRilGJUYrRiVGJUYtRi9GKUYlRiUtRjI2JEYuRixGJUYsRi9GJSouRiRGJUYpRiUsJiokRitGJUYlRiVGJUYlLUkiQkdGKkYzRiVGLkYlRitGL0YlKixGJEYlRi5GJUYpRiVJJEIwMUdGKkYlRixGL0YvKiYjRiYiIiRGJUY4RiVGJQ==

collect(convert(series(convert(series(eqL,x,4),polynom),y,4),polynom),[x,y],distributed);

LDQqJiwoKigjIiIiIiInRidJInRHNiJGJy0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYmRidGJ0YnIiIjNiNJIkxHRio2JCIiIUY3RidGJyomI0YnRjNGJy0tJkYuNiRGJ0YnRjRGNkYnISIiKiZGKUYnLS0mRi42JEYnRjNGNEY2RidGJ0YnKUkieEdGKiIiJEYnRicqKCwsKigjRiciIzdGJ0YpRictLSZGLjYnRidGJ0YnRjNGM0Y0RjZGJ0YnKihGJkYnRilGJy0tJkYuNiVGJ0YnRidGNEY2RidGJyomRjlGJy0tJkYuNiVGJ0YnRjNGNEY2RidGPiooRjlGJ0YpRictLSZGLjYlRidGM0YzRjRGNkYnRicqJkYpRictLSZGLjYjRidGNEY2RidGJ0YnSSJ5R0YqRidGREYnRicqJiwqKihGOUYnRilGJ0ZWRidGJ0ZqbkY+KiZGKUYnLS0mRi42I0YzRjRGNkYnRicjRjNGRkYnRicpRkVGM0YnRicqKCwsKigjRiciIiVGJ0YpRictLSZGLjYmRidGJ0YzRjNGNEY2RidGJyooRjlGJ0YpRidGOkYnRidGQEY+KihGOUYnRilGJy0tJkYuNiRGM0YzRjRGNkYnRicqJkYpRictRjVGNkYnRidGJ0Zeb0YnRmhvRidGJyomLCoqJkY5RidGaXBGJ0Y+Rj9GJyooRlxwRidGKUYnLS1GZnA2I0kiQkdGKkY2RidGJyooRjlGJ0YpRictRmFxRjZGJ0YnRidGRUYnRicqLEY5RidGKUYnRmNxRidGRUYnKUZeb0YzRj5GJyooLCwqJkY5RidGY29GJ0Y+RlpGJ0ZpbkYnKihGSkYnRilGJy0tJkYuNiVGM0YzRjNGYHFGNkYnRicqKEY5RidGKUYnLS1GZW9GYHFGNkYnRidGJ0Zeb0YnRkVGJ0YnKigsJkZeckYnKihGOUYnRilGJ0kkQjAxR0YqRidGPkYnRl5vRj5GRUYnRicqKEYpRidGaXBGJ0Zeb0Y+Rj4=

expand(solve(coeff(coeff(%,x,1),y,1),(D[1, 2,2](L))(0, 0)));

LCotLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIzYjSSJCR0YqNiQiIiFGMCEiIiomRiwiIiItLSZGJjYjRjM2I0kiTEdGKkYvRjNGMSomI0YzIiInRjMtLSZGJjYlRixGLEYsRi1GL0YzRjEqJkkidEdGKkYxLS1GJUY4Ri9GM0Yz

L12Zm:=factor(subs(Tm=Zm^2,1/2*(-B01Zm-2*L10Zm-B03Zm+L01ftTm/t2Tm)));

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNHRicpRikiIidGJ0YnKiYiI1VGJylGKSIiJkYnRiwqJiIkRSJGJylGKSIiJUYnRicqJiIkQSNGJ0YoRidGLComIiRrIkYnKUYpIiIjRidGLComIiRRI0YnRilGJ0YsIiQ+IkYsRicsJkYpRidGJ0YnRiwpLCYqJEZFRidGJ0YmRidGOEYsRiw=

Cm22sol:=L12Zm+1/3*Q02;

LCYqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNHRicpRikiIidGJ0YnKiYiI1VGJylGKSIiJkYnRiwqJiIkRSJGJylGKSIiJUYnRicqJiIkQSNGJ0YoRidGLComIiRrIkYnKUYpIiIjRidGLComIiRRI0YnRilGJ0YsIiQ+IkYsRicsJkYpRidGJ0YnRiwpLCYqJEZFRidGJ0YmRidGOEYsRiwqJiNGJ0YmRidJJFEwMkdGKkYnRic=

n:=12: series(subs(Zm=Zmser(n),Q02=coeff(coeff(convert(Qser(n),polynom),x,0),y,2),Cm22sol),t,n+1);

Ky9JInRHNiIiIiNGJSIjPSIiJSIkIj0iIiciJSopPiIiKSImcksjIiM1LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNw==

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

LC4qJiInalxHIiIiKUkidEc2IiIjN0YlRiUqJiImcksjRiUpRiciIzVGJUYlKiYiJSopPkYlKUYnIiIpRiVGJSomIiQiPUYlKUYnIiInRiVGJSomIiM9RiUpRiciIiVGJUYlKiYiIiNGJSlGJ0Y7RiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-3,1): the series L21</Text-field>

collect(convert(series(convert(series(eqL,x,4),polynom),y,4),polynom),[x,y],distributed);

expand(solve(coeff(coeff(%,x,2),y,0),(D[1, 1,2](L))(0, 0)));

LCgqJiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiQ2I0kiTEdGLTYkIiIhRjJGJSEiIiooRiRGJUkidEdGLUYzLS0mRik2I0YlRi9GMUYlRiUqJiMiIiUiIiRGJUY1RjNGMw==

L21ftZm:=factor(subs(Tm=Zm^2,-L01ftTm+L10Zm-2/3));

LCQqLCMiIiMiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMCokKUYpIiInRidGJyomIiM5RicpRikiIiZGJ0YrKiYiI1ZGJylGKSIiJUYnRicqJiIjR0YnKUYpRiZGJ0YrKiYiI2RGJylGKUYlRidGKyomRi9GJ0YpRidGK0YmRitGJywmRilGJ0YnRidGKyksJiokRj1GJ0YnRiZGJ0YmRitGKw==

Cm31ftsol:=L21ftZm+t/3*Q11;

LCYqLCMiIiMiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMCokKUYpIiInRidGJyomIiM5RicpRikiIiZGJ0YrKiYiI1ZGJylGKSIiJUYnRicqJiIjR0YnKUYpRiZGJ0YrKiYiI2RGJylGKUYlRidGKyomRi9GJ0YpRidGK0YmRitGJywmRilGJ0YnRidGKyksJiokRj1GJ0YnRiZGJ0YmRitGKyooI0YnRiZGJ0kidEdGKkYnSSRRMTFHRipGJ0Yn

n:=12: series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),Cm31ftsol),t,n+1);

Ky9JInRHNiIiIiIiIiMiIiciIiUiI11GJyIkI1wiIikiJV1gIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

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

LC4qJiImd0AnIiIiKUkidEc2IiIjNkYlRiUqJiIlXWBGJSlGJyIiKkYlRiUqJiIkI1xGJSlGJyIiKEYlRiUqJiIjXUYlKUYnIiImRiVGJSomIiInRiUpRiciIiRGJUYlRidGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-1) and (3,-1) and (5,-1): the series B10, B30, and B50</Text-field>

algBx0ftxTm:=(x^2+1)*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*Bx0ftx^3+(x^2+2)*(Tm^4*x^4+8*Tm^3*x^4+2*Tm^4*x^2+18*Tm^2*x^4-48*Tm^3*x^2+Tm^4+36*Tm^2*x^2-27*x^4+8*Tm^3+18*Tm^2-54*x^2-27)*Bx0ftx^2+(2*Tm^4*x^4+3*Tm^4*x^2+40*Tm^2*x^4-56*Tm^3*x^2-24*Tm*x^4+Tm^4+58*Tm^2*x^2-18*x^4+8*Tm^3-24*Tm*x^2+18*Tm^2-45*x^2-27)*Bx0ftx+x^2*(Tm-1)^2*(Tm^2+4*x^2-6*Tm+5);

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

ser:=map(factor,op(2,algeqtoseries(algBx0ftxTm,x,Bx0ftx,5,true)));

B10ftTm:=coeff(ser,x,2); B30ftTm:=coeff(ser,x,4); B50ftTm:=coeff(ser,x,6);

LCQqKCwmSSNUbUc2IiIiIiIiJiEiIkYnKSwmRiVGJ0YnRikiIiNGJyksJkYlRiciIiRGJ0YvRilGKQ==

KigsKiokKUkjVG1HNiIiIiQiIiJGKSomIiNWRikpRiYiIiNGKSEiIiomIiRKIkYpRiZGKUYpIiRuIkYpRikpLCZGJkYpRilGLkYoRikpLCZGJkYpRihGKSIiJ0Yu

LCQqKCwuKiQpSSNUbUc2IiIiJiIiIkYqKiYiJDgiRiopRiciIiVGKiEiIiomIiVRPEYqKUYnIiIkRipGKiomIiUhcCNGKilGJyIiI0YqRi8qJiIlciQqRipGJ0YqRi8iJURcRi9GKiksJkYnRipGKkYvRi5GKiksJkYnRipGM0YqIiIqRi9GLw==

C1m1ftsol:=B10ftTm; C3m1ftsol:=B30ftTm+t/3*Q21; C5m1ftsol:=B50ftTm+t/3*Q41;

LCQqKCwmSSNUbUc2IiIiIiIiJiEiIkYnKSwmRiVGJ0YnRikiIiNGJyksJkYlRiciIiRGJ0YvRilGKQ==

LCYqKCwqKiQpSSNUbUc2IiIiJCIiIkYqKiYiI1ZGKilGJyIiI0YqISIiKiYiJEoiRipGJ0YqRioiJG4iRipGKiksJkYnRipGKkYvRilGKiksJkYnRipGKUYqIiInRi9GKiooI0YqRilGKkkidEdGKEYqSSRRMjFHRihGKkYq

LCYqKCwuKiQpSSNUbUc2IiIiJiIiIkYqKiYiJDgiRiopRiciIiVGKiEiIiomIiVRPEYqKUYnIiIkRipGKiomIiUhcCNGKilGJyIiI0YqRi8qJiIlciQqRipGJ0YqRi8iJURcRi9GKiksJkYnRipGKkYvRi5GKiksJkYnRipGM0YqIiIqRi9GLyooI0YqRjNGKkkidEdGKEYqSSRRNDFHRihGKkYq

n:=12: normal(series(subs(Tm=Tmser(n),C1m1ftsol),t,n+1));

Ky9JInRHNiIiIiIiIiUiIzkiIiciJCE9IiIpIiVDQiIjNSImRTAkIiM3LUkiT0clKnByb3RlY3RlZEc2I0YlRic=

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

LCwqJiImRTAkIiIiKUkidEc2IiIjNkYlRiUqJiIlQ0JGJSlGJyIiKkYlRiUqJiIkIT1GJSlGJyIiKEYlRiUqJiIjOUYlKUYnIiImRiVGJSokKUYnIiIkRiVGJQ==

n:=12: normal(series(subs(Tm=Tmser(n),Q21=coeff(coeff(convert(Qser(n),polynom),x,2),y,1),C3m1ftsol),t,n+1));

Ky1JInRHNiIiIiUiIiciIygpIiIpIiVnOSIjNSImaUQjIiM3LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOQ==

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

LCoqJiImaUQjIiIiKUkidEc2IiIjNkYlRiUqJiIlZzlGJSlGJyIiKkYlRiUqJiIjKClGJSlGJyIiKEYlRiUqJiIiJUYlKUYnIiImRiVGJQ==

n:=16: normal(series(subs(Tm=Tmser(n),Q41=coeff(coeff(convert(Qser(n),polynom),x,4),y,1),C5m1ftsol),t,n+1));

Ky9JInRHNiIiIzoiIikiJFclIiM1IiUwIyoiIzciJ11nOyIjOSIoQGd6IyIjOy1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIz0=

coeff(coeff(convert(Cser(n),polynom),x,5),y,-1);

LCwqJiIoQGd6IyIiIilJInRHNiIiIzpGJUYlKiYiJ11nO0YlKUYnIiM4RiVGJSomIiUwIypGJSlGJyIjNkYlRiUqJiIkVyVGJSlGJyIiKkYlRiUqJkYpRiUpRiciIihGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,-3): the series B12</Text-field>

collect(map(Z->convert(series(Z,y,3),polynom),convert(series(eqB,x,3),polynom)),[x,y],distributed);

LCwqLCMiIiIiIiNGJUkidEc2IkYlLUkiTEdGKDYkIiIhRixGJUkieUdGKEYlKUkieEdGKEYmISIiRiUqKCwmKihGJEYlRidGJS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRiU2I0YqRitGJUYlKihGJEYlRidGJUkkTDEwR0YoRiVGMEYlRi9GMEYtRiVGJSomLCoqKCNGJSIiJUYlRidGJS0tJkY3NiRGJUYlRjxGK0YlRiUqKEYkRiVGJ0YlRilGJUYlKiZGJEYlLUkiQkdGKEYrRiVGMComRidGJS0tJkY3NiRGJUYmNiNGS0YrRiVGJUYlRi1GJUYlKihGJ0YlRkpGJUYvRjBGMComLCoqKEYkRiVGJ0YlLS0mRjc2JUYlRiZGJkZRRitGJUYlKiZGJ0YlLS1GNkZRRitGJUYlLS0mRjc2I0YmRlFGK0YwI0YlIiIkRjBGJSlGLUYmRiVGJQ==

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

LCoqKCMiIiIiIiNGJUkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiVGJUYmRiY2I0kiQkdGKDYkIiIhRjRGJUYlKiZGJ0YlLS0mRiw2I0YlRjFGM0YlRiUtLSZGLDYjRiZGMUYzISIiI0YlIiIkRj4=

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

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlIiIiIiIjRi02I0kiQkdGKjYkIiIhRjEsKComRi1GLC0tJkYmNiNGLEYuRjBGLCEiIiooRi1GLEkidEdGKkY4LS0mRiY2I0YtRi5GMEYsRiwqJiNGLSIiJEYsRjpGOEYs

B12ftZm:=factor(subs(Tm=Zm^2,-B10ftTm+B01Zm+1/3));

LCQqLCMiIiUiIiQiIiIsJkkjWm1HNiJGJ0YnISIiRicsMCokKUYpIiInRidGJyomIiM5RicpRikiIiZGJ0YrKiYiI0pGJylGKUYlRidGJyomRiVGJylGKUYmRidGKyomIiNARicpRikiIiNGJ0YrKiZGL0YnRilGJ0YrRiZGK0YnLCZGKUYnRidGJ0YrKSwmKiRGO0YnRidGJkYnRiZGK0Yn

C1m3ftsol:=B12ftZm+t/3*Q11:

n:=16: normal(series(subs(Zm=Zmser(n),Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),C1m3ftsol),t,n+1));

Ky9JInRHNiIiIiMiIiciI1UiIikiJCpvIiM1IiYjWzUiIzciJ0VdOiIjOS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

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

LC4qJiIoSyVvQSIiIilJInRHNiIiIzpGJUYlKiYiJ0VdOkYlKUYnIiM4RiVGJSomIiYjWzVGJSlGJyIjNkYlRiUqJiIkKm9GJSlGJyIiKkYlRiUqJiIjVUYlKUYnIiIoRiVGJSomIiIjRiUpRiciIiZGJUYl

<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(eqB,x,4),polynom)),[x,y],distributed,factor);

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

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

LCwqKCMiIiIiIzdGJUkidEc2IkYlLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiVGJUYlRiU2I0kiTEdGKDYkIiIhRjRGJUYlKigjRiUiIiNGJUYnRiUtLSZGLDYjRiVGMUYzRiVGJSomRjZGJS0tRjo2I0kiQkdGKEYzRiUhIiIqKEY2RiVGJ0YlLS0mRiw2JUYlRiVGN0Y/RjNGJUYlKiZGJ0YlLS0mRiw2I0Y3Rj9GM0YlRiU=

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

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlIiIiRiwiIiM2I0kiQkdGKjYkIiIhRjEsKiomI0YsIiInRiwtLSZGJjYlRixGLEYsNiNJIkxHRipGMEYsISIiLS0mRiY2I0YsRjpGMEY8KiZJInRHRipGPC0tRj9GLkYwRixGLComRi1GLC0tJkYmNiNGLUYuRjBGLEY8

B21Zm:=factor(subs(Tm=Zm^2,(-L30Zm-L10Zm+B10ftTm/t2Tm-2*B01Zm )/2));

LCQqLiMiI2siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDIqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLCokKUYpIiInRidGJyomIiMkKkYnKUYpIiIlRidGJyomIiNVRidGKEYnRiwqJkY0RicpRikiIiNGJ0YnKiYiI3dGJ0YpRidGLCIjUUYsRicsJkYpRidGJ0YnRiwpLCYqJEY/RidGJ0YmRidGN0YsRic=

C2m2sol:=B21Zm+1/3*Q20:

n:=16: normal(series(subs(Zm=Zmser(n),Q20=coeff(coeff(convert(Qser(n),polynom),x,2),y,0),C2m2sol),t,n+1));

KzFJInRHNiIiIiIiIiUiI0AiIiciJFIkIiIpIiVzXSIjNSImOVIoIiM3IihZejEiIiM5LUkiT0clKnByb3RlY3RlZEc2I0YlIiM7

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

LDAqJiIpVS5UOiIiIilJInRHNiIiIztGJUYlKiYiKFl6MSJGJSlGJyIjOUYlRiUqJiImOVIoRiUpRiciIzdGJUYlKiYiJXNdRiUpRiciIzVGJUYlKiYiJFIkRiUpRiciIilGJUYlKiYiI0BGJSlGJyIiJ0YlRiUqJClGJyIiJUYlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0): the series P00</Text-field>

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

LCYtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiIjYjSSJMR0YqNiQiIiFGMEYsLS0mRiY2IyIiIzYjSSJCR0YqRi9GLA==

already found when studying walks starting from (0,0) and ending at (-2,0)

P00Zm:=factor(L10Zm+B01Zm);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYw

C00sol:=P00Zm-1/3*Q00;

LCYqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYwKiYjRidGJkYnSSRRMDBHRipGJ0Yw

n:=12: normal(series(subs(Zm=Zmser(n),Q00=coeff(coeff(convert(Qser(n),polynom),x,0),y,0),C00sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIioiIiUiIyMqIiInIiVJNSIiKSImbUEiIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlIiM3

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

LC4qJiInT0U6IiIiKUkidEc2IiIjN0YlRiUqJiImbUEiRiUpRiciIzVGJUYlKiYiJUk1RiUpRiciIilGJUYlKiYiIyMqRiUpRiciIidGJUYlKiYiIipGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (2,0): the series P20</Text-field>

series(coeff(series(Pexpr,x,5),x,2),y,2);

KydJInlHNiIsJiomIyIiIiIiJ0YoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiVGKEYoRig2I0kiTEdGJDYkIiIhRjVGKEYoKiYjRigiIiNGKC0tJkYtNiVGKEYoRjg2I0kiQkdGJEY0RihGKEY1LUkiT0dGLzYjRihGKA==

P20Zm:=factor(L30Zm+B21Zm);

LCQqMCMiI0siIiQiIiIsJkkjWm1HNiJGJ0YnRidGJywmKiQpRikiIiNGJ0YnIiImISIiRicsKiokKUYpIiIlRidGJyomIiM3RicpRilGJkYnRjAqJkYuRidGLUYnRidGJ0YnRidGN0YnLCZGKUYnRidGMEYnKSwmRixGJ0YmRiciIidGMEYn

C20sol:=P20Zm-1/3*Q20:

n:=16: normal(series(subs(Zm=Zmser(n),Q20=coeff(coeff(convert(Qser(n),polynom),x,2),y,0),C20sol),t,n+1));

KzFJInRHNiIiIiQiIiUiI14iIiciJEwoIiIpIiZ2LCIiIzUiJ0cxOSIjNyIoeEUmPiIjOS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzs=

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

LDAqJiIpMVNKRiIiIilJInRHNiIiIztGJUYlKiYiKHhFJj5GJSlGJyIjOUYlRiUqJiInRzE5RiUpRiciIzdGJUYlKiYiJnYsIkYlKUYnIiM1RiVGJSomIiRMKEYlKUYnIiIpRiVGJSomIiNeRiUpRiciIidGJUYlKiYiIiRGJSlGJyIiJUYlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,2): the series P02</Text-field>

series(Pexpr,x,2);

KylJInhHNiIsKComLUkiQkdGJDYkIiIhRioiIiJJInlHRiQhIiJGLS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRis2I0kiTEdGJDYkRipGLEYrKiYtRihGOEYrRixGLUYrRiosKComLS1GMDYjRihGKUYrRixGLUYtKiYjRisiIiNGKy0tJkYxNiRGK0YrRjZGOEYrRisqJi1GPkY4RitGLEYtRitGKy1JIk9HRjNGNUZC

series(coeff(series(Pexpr,x,2),x,0),y,4);

KytJInlHNiIsJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiNiNJIkxHRiQ2JCIiIUYyRi4tLSZGKTYjIiIjNiNJIkJHRiRGMUYuRjIsJi0tJkYpNiRGLkY3Ri9GMUYuKiYjRi5GN0YuLS0mRik2JEY3RjdGOEYxRi5GLkYuLCYqJkZARi4tLSZGKTYlRi5GN0Y3Ri9GMUYuRi4qJiNGLiIiJ0YuLS0mRik2JUY3RjdGN0Y4RjFGLkYuRjctSSJPR0YrRi0iIiQ=

P02Zm:=factor(L12Zm+B03Zm);

LCQqLiMiI0siIiQiIiIpSSNabUc2IkYmRicsJkYpRidGJyEiIkYnLDQqJClGKSIiKUYnRicqJiIjNUYnKUYpIiIoRidGLComIiNBRicpRikiIidGJ0YnKiYiI2FGJylGKSIiJkYnRiwqJiIjZ0YnKUYpIiIlRidGJyomIiNVRidGKEYnRicqJiIkQSRGJylGKSIiI0YnRicqJiIjJylGJ0YpRidGJyIjVkYnRicsJkYpRidGJ0YnRiwpLCYqJEZFRidGJ0YmRidGOEYsRic=

C02sol:=P02Zm-1/3*Q02:

n:=12: normal(series(subs(Zm=Zmser(n),Q02=coeff(coeff(convert(Qser(n),polynom),x,0),y,2),C02sol),t,n+1));

Ky9JInRHNiIiIiIiIiMiIzciIiUiJFEiIiInIiVXOyIiKSImQS4jIiM1LUkiT0clKnByb3RlY3RlZEc2I0YlRic=

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

LC4qJiInKyJmIyIiIilJInRHNiIiIzdGJUYlKiYiJkEuI0YlKUYnIiM1RiVGJSomIiVXO0YlKUYnIiIpRiVGJSomIiRRIkYlKUYnIiInRiVGJSomRilGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (1,1): the series P11</Text-field>

series(Pexpr,x,2);;

series(coeff(series(Pexpr,x,2),x,1),y,3);

KylJInlHNiIsJiomIyIiIiIiI0YoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRGKEYoNiNJIkxHRiQ2JCIiIUY1RihGKC0tJkYtNiRGKEYpNiNJIkJHRiRGNEYoRjUsJiomRidGKC0tJkYtNiVGKEYoRilGMkY0RihGKComRidGKC0tJkYtNiVGKEYpRilGOkY0RihGKEYoLUkiT0dGLzYjRihGKQ==

P11ftZm:=factor(L21ftZm+B12ftZm);

C11ftsol:=P11ftZm-t/3*Q11:

n:=16: normal(series(subs(Zm=Zmser(n), Q11=coeff(coeff(convert(Qser(n),polynom),x,1),y,1),C11ftsol),t,n+1));

KzFJInRHNiIiIiMiIiUiI0YiIiciJFEkIiIpIiV2VSIjNSImTV8mIiM3IiczJ0coIiM5LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjOw==

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

LDAqJiIoR3d5KiIiIilJInRHNiIiIzpGJUYlKiYiJzMnRyhGJSlGJyIjOEYlRiUqJiImTV8mRiUpRiciIzZGJUYlKiYiJXZVRiUpRiciIipGJUYlKiYiJFEkRiUpRiciIihGJUYlKiYiI0ZGJSlGJyIiJkYlRiUqJiIiI0YlKUYnIiIkRiVGJQ==

<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:

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

with(LREtools):

L10Zm:=(32/3)*Zm^3*(Zm^3-3*Zm^2+5*Zm+5)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJiIiJkYnRilGJ0YnRjJGJ0YnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYn

algL10:=factor(subs(t=sqrt(s),factor(resultant(numer(L10Zm-L10),algZm,Zm)))):

algeqtodiffeq(algL10,L10(s));

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

de:=%:

rec:=diffeqtorec(de,L10(s),co(k)):

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

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

PCUvLUkjY29HNiI2IyIiISMiIiMiIiQvLUYlNiMiIiIjIiImRisvLUYlNiNGKiMiI0tGKw==

HG solution

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

LCYqNiMiIiUiI0YiIiIpIiIkI0YnIiIjRicpSSNQaUclKnByb3RlY3RlZEdGKiEiIi1JJkdBTU1BRzYkRi5JKF9zeXNsaWJHNiI2IyNGK0YpRi8tRjE2IyMiIiYiIidGJykiIztJImtHRjRGJy1GMTYjLCZGPkYnRipGJ0YnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RicjRiVGKUYnRi8tRjE2IywmRj5GJ0YrRidGL0YnKjQjIiIpRiZGJ0YsRi9GN0YvRjBGJ0Y8RidGP0YnLUYxNiMsJkY+RidGOUYnRictRjE2IywmRj5GJyNGOkYpRidGL0ZLRi9GJw==

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),L10Zm),s,n/2+1)):

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

PCUsKComLC4qJiIlOyMqIiIiKUkia0c2IiIiJkYoRigqJiImK3cmRigpRioiIiVGKEYoKiYiJyFvTiJGKClGKiIiJEYoRigqJiInKy86RigpRioiIiNGKEYoKiYiJlcieUYoRipGKEYoIiY/XiJGKEYoLUkjY29HRis2I0YqRihGKComLC4qJiIlXzZGKEYpRighIiIqJiImITM1RihGL0YoRkMqJiImISlbJEYoRjNGKEZDKiYiJlMnZkYoRjdGKEZDKiYiJm8uJkYoRipGKEZDIiYrbyJGQ0YoLUY9NiMsJkYqRihGKEYoRihGKComLC4qJiIjT0YoRilGKEYoKiYiJDAlRihGL0YoRigqJiIlIXoiRihGM0YoRigqJiIldlFGKEY3RihGKComIiUlNCVGKEYqRihGKCIlIW8iRihGKC1GPTYjLCZGKkYoRjhGKEYoRigvLUY9NiMiIiEjRjhGNC8tRj02I0YoI0YsRjQ=

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

LCgqMCIjOyIiIiwmKiYiIidGJUkia0c2IkYlRiUiIiZGJUYlLCYqJiIiI0YlRilGJUYlIiIkRiVGJSwmRi1GJUYlRiVGJSwmRidGJSIiKEYlRiUsJiomIiIlRiVGKUYlRiUiIipGJUYlLUkjY29HRio2I0YpRiVGJSouIiIpRiVGLEYlLCZGNEYlRjJGJUYlLCZGKUYlRi5GJUYlLCgqJiIjPUYlKUYpRi5GJUYlKiYiI2pGJUYpRiVGJSIjXUYlRiUtRjg2IywmRilGJUYlRiVGJSEiIiouLCZGNEYlRitGJUYlLCYqJkYvRiVGKUYlRiVGMkYlRiUsJkYpRiVGL0YlRiVGPUYlLCZGTEYlRjtGJUYlLUY4NiNGPUYlRiU=

(hypergeomsols(rec1,co(k),{co(0) = 2/3, co(1) = 5/3}));

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

seq((4/27)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+7/6)*GAMMA(k+1/2)/(sqrt(Pi)*GAMMA(2/3)*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);

NiwjIiIjIiIqRiNGI0YjRiNGI0YjRiNGI0Yj

seq((8/27)*GAMMA(2/3)*16^k*GAMMA(k+5/6)*GAMMA(k+1/2)/(sqrt(Pi)*GAMMA(5/6)*GAMMA(k+5/3)*GAMMA(k+2))/ (fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k)*16^k), k=1..10);

NiwjIiIlIiIqRiNGI0YjRiNGI0YjRiNGI0Yj

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

NiwtSSZRY29uakc2IjYlIiIhRiciIiMsJComIyIiIiIiJUYsLUYkNiVGJ0YnRi1GLEYsLCQqJiNGLCIjREYsLUYkNiVGJ0YnIiInRixGLCwkKiYjRiwiJCc+RiwtRiQ2JUYnRiciIilGLEYsLCQqJiNGLCIlazxGLC1GJDYlRidGJyIjNUYsRiwsJComI0YsIiZDdSJGLC1GJDYlRidGJyIjN0YsRiwsJComI0YsIidUUz1GLC1GJDYlRidGJyIjOUYsRiwsJComI0YsIigrXC8jRiwtRiQ2JUYnRiciIztGLEYsLCQqJiNGLCIpVyFSTyNGLC1GJDYlRidGJyIjPUYsRiwsJComI0YsIio7YzUjR0YsLUYkNiVGJ0YnIiM/RixGLA==

paper version

seq(16^k/9*(2*fact(1/2,k)*fact(7/6,k)/fact(2,k)/fact(4/3,k)+4*fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k) +3*fact(1/2,k)^2/fact(2,k)^2),k=0..10);

Ni0iIiIiIiMiIzciJCwiIiUvNSImPDUiIidfIUgiIih1TGUiIilzITQsIyIqXWJIaSMiK1NWcyVcJA==

coeff(coeff(convert(Cser(20),polynom),x,-2),y,0);

LDgqJiIrU1ZzJVwkIiIiKUkidEc2IiIjP0YlRiUqJiIqXWJIaSNGJSlGJyIjPUYlRiUqJiIpcyE0LCNGJSlGJyIjO0YlRiUqJiIodUxlIkYlKUYnIiM5RiVGJSomIidfIUgiRiUpRiciIzdGJUYlKiYiJjw1IkYlKUYnIiM1RiVGJSomIiUvNUYlKUYnIiIpRiVGJSomIiQsIkYlKUYnIiInRiVGJSomRjlGJSlGJyIiJUYlRiUqJiIiI0YlKUYnRkpGJUYlRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-4,0) : HG</Text-field>

L30Zm:=-(32/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+28*Zm^6-42*Zm^5+78*Zm^4-126*Zm^3+28*Zm^2-142*Zm-71)/((Zm+1)*(Zm^2+3)^6);

algL30:=factor(subs(t=sqrt(s),factor(resultant(numer(L30-L30Zm),algZm,Zm)))):

de0:=algeqtodiffeq(algL30,L30(s)); nops(%);

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

IiM2

de:=de0:

rec:=diffeqtorec(de,L30(s),co(k)):

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

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

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

PCQvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiIyIiJA==

HG solution

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

LCYqOCMiIiUiJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJykiIztJImtHRjFGJy1GLTYjLCZGOEYnRipGJ0YnLUYtNiMsJkY4RicjIiIoRjVGJ0YnLCgqJiIjQEYnKUY4RitGJ0YnKiYiI0lGJ0Y4RidGJ0Y3RidGJylJI1BpR0YvRiohIiItRi02IyNGK0YpRkktRi02IywmRjhGJ0YpRidGSS1GLTYjLCZGOEYnI0ZARilGJ0ZJRicqNiMiIilGJkYnRkpGJ0Y2RidGOUYnLUYtNiMsJkY4RidGM0YnRicsKComIiNSRidGREYnRicqJiIjbUYnRjhGJ0YnIiM1RklGJ0YsRklGR0ZJRk1GSS1GLTYjLCZGOEYnI0ZWRilGJ0ZJRic=

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

seq((4/729)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+1/2)*GAMMA(k+7/6)*(21*k^2+30*k+16)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+3)*GAMMA(k+7/3))/ ((21*k^2+30*k+16)*fact(1/2,k)*fact(7/6,k)*16^k/fact(2,k+1)/fact(4/3,k+1)),k=1..10);

NiwjIiIjIiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

seq((8/729)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)*(39*k^2+66*k-10)/(GAMMA(5/6)*sqrt(Pi)*GAMMA(k+3)*GAMMA(k+8/3))/ ((39*k^2+66*k-10)*fact(1/2,k)*fact(5/6,k)/fact(2,k+1)/fact(5/3,k+1)*16^k), k=1..10);

NiwjIiIlIiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

seq(Qconj(2,0,2*k)/(3*16^k*k*fact(1/2,k)^2/fact(2,k)/fact(2,k+1)),k=1..10);

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

paper version

seq(16^k/3^5* (2*(21*k^2+30*k+16)*fact(1/2,k)*fact(7/6,k)/fact(2,k+1)/fact(4/3,k+1) +4*(39*k^2+66*k-10)*fact(1/2,k)*fact(5/6,k)/fact(2,k+1)/fact(5/3,k+1) +3^5*k*fact(1/2,k)^2/fact(2,k)/fact(2,k+1)) ,k=0..10);

Ni0iIiEiIiIiIikiI3YiJCd5IiUkKSkpIic9ZjUiKGdUSiIiKUM5Im8iIip2aks/IyIrKUhvWSVI

coeff(coeff(convert(Cser(20),polynom),x,-4),y,0);

LDYqJiIrKUhvWSVIIiIiKUkidEc2IiIjP0YlRiUqJiIqdmpLPyNGJSlGJyIjPUYlRiUqJiIpQzkibyJGJSlGJyIjO0YlRiUqJiIoZ1RKIkYlKUYnIiM5RiVGJSomIic9ZjVGJSlGJyIjN0YlRiUqJiIlJCkpKUYlKUYnIiM1RiVGJSomIiQneUYlKUYnIiIpRiVGJSomIiN2RiUpRiciIidGJUYlKiZGQUYlKUYnIiIlRiVGJSokKUYnIiIjRiVGJQ==

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),L30Zm),s,n/2+1)):

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

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

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

LCgqMCIjOyIiIiwmKiYiIidGJUkia0c2IkYlRiUiIiZGJUYlLCYqJiIiI0YlRilGJUYlIiIkRiVGJSwmRi1GJUYlRiVGJSwmRidGJSIiKEYlRiUsLiomIiRrJEYlKUYpRitGJUYlKiYiJWxXRiUpRikiIiVGJUYlKiYiJl0sI0YlKUYpRi9GJUYlKiYiJmwtJUYlKUYpRi5GJUYlKiYiJic+S0YlRilGJUYlIiU/WUYlRiUtSSNjb0dGKjYjRilGJUYlKi4iIilGJUYsRiUsJkYpRiVGL0YlRiUsJkYpRiVGLkYlRiUsMComIiVfbEYlKUYpRihGJUYlKiYiJnEuKUYlRjZGJUYlKiYiJ2JQT0YlRjlGJUYlKiYiJzAjPShGJUY9RiVGJSomIiczPltGJUZARiVGJSomIidJQ0BGJUYpRiUhIiIiJytzRkZZRiUtRkU2IywmRilGJUYlRiVGJUZZKi4sJiomRi9GJUYpRiVGJSIjNUYlRiUsJkYpRiVGOkYlRiVGSUYlLCZGam5GJSIjNkYlRiUsLkY0RiUqJiIlWEVGJUY5RiVGJSomIiVJZkYlRj1GJUYlKiYiJWxIRiVGQEYlRiUqJiIlQ1JGJUYpRiVGWSIlZ0xGWUYlLUZFNiNGSkYlRiU=

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

LCYqOCMiI2siJEgoIiIiKSIiJCNGJyIiI0YnKUkjUGlHJSpwcm90ZWN0ZWRHRiohIiItSSZHQU1NQUc2JEYuSShfc3lzbGliRzYiNiMjRitGKUYvLUYxNiMjIiImIiInRicpIiM7SSJrR0Y0RictRjE2IywmRj5GJ0YqRidGJy1GMTYjLCZGPkYnIyIiKEY7RidGJy1GMTYjLCZGPkYnRilGJ0YvLUYxNiMsJkY+RicjRkZGKUYnRi8sKComIyIjQEY9RicpRj5GK0YnRicqJiMiIzoiIilGJ0Y+RidGJ0YnRidGJ0YnKjYjIiMhKUYmRidGN0YvRjBGJ0YsRi9GPEYnRj9GJy1GMTYjLCZGPkYnRjlGJ0YnRkdGLy1GMTYjLCZGPkYnI0ZWRilGJ0YvLCgqJiMiI1IiIzVGJ0ZSRidGLyomIyIjTEY6RidGPkYnRi9GJ0YnRidGLw==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-6,0) : HG</Text-field>

L50Zm := (32/3)*Zm^3*(Zm^13-17*Zm^12+68*Zm^11-104*Zm^10+381*Zm^9-1177*Zm^8+1392*Zm^7-2712*Zm^6+4491*Zm^5-1299*Zm^4+7500*Zm^3+1728*Zm^2+4599*Zm+1533)*(Zm-1)^2/((Zm+1)*(Zm^2+3)^9);

algL50:=factor(subs(t=sqrt(s),factor(resultant(numer(L50-L50Zm),algZm,Zm)))):

de0:=algeqtodiffeq(algL50,L50(s)); nops(%);

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

IiM4

de:=de0:

rec:=diffeqtorec(de,L50(s),co(k)):

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

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

with(LREtools):

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

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMjIiIlIiIkLy1GJTYjRjMjIiNsRjMvLUYlNiNGMiMiJCcqKUYzLy1GJTYjIiImIiUmKlIvLUYlNiMiIiciJk9NJi8tRiU2IyIiKCInWDJzLy1GJTYjIiIpIyIpJVxyJUhGMy8tRiU2IyIiKiIqamdKTiI=

HG solution

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

LCYqOCMiIiUiJiRvPiIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRicpIiM7SSJrR0YxRictRi02IywmRjhGJ0YqRidGJy1GLTYjLCZGOEYnIyIiKEY1RidGJywsKiYiJCpbRicpRjhGJUYnRicqJiIlKVIiRicpRjhGKUYnRicqJiIkRipGJylGOEYrRidGJyomIiRBI0YnRjhGJyEiIiIlcz9GTUYnKUkjUGlHRi9GKkZNLUYtNiMjRitGKUZNLUYtNiMsJkY4RicjIiM1RilGJ0ZNLUYtNiMsJkY4RidGJUYnRk1GJyo2IyIiKUYmRidGUUYnRjZGJ0Y5RictRi02IywmRjhGJ0YzRidGJywsKiYiJWo3RidGREYnRicqJiIlMFZGJ0ZHRidGJyomIiRrJkYnRkpGJ0ZNKiYiJWNxRidGOEYnRk0iJSFbIkYnRidGLEZNRk9GTS1GLTYjLCZGOEYnIyIjNkYpRidGTUZZRk1GJw==

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

In fact there seems to be a shorter recurrence relation

n:=100: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),L50Zm),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)= 4/3}));

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

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (-8,0) : HG</Text-field>

L70Zm;

algL70:=factor(subs(t=sqrt(s),factor(resultant(numer(L70-L70Zm),algZm,Zm)))):

de0:=algeqtodiffeq(algL70,L70(s)); nops(%);

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

IiM6

de:=de0:

rec:=diffeqtorec(de,L70(s),co(k)):

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

with(LREtools):

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

PC8vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCMiIzVGNC8tRiU2IyIiJSMiJFEjRjQvLUYlNiMiIiYiJSs5Ly1GJTYjIiInIiZ5QSMvLUYlNiMiIigiJycqKlEkLy1GJTYjIiIpIyIpT006OkY0Ly1GJTYjIiIqIilQTl91Ly1GJTYjRjYiK01bIVw0Ii8tRiU2IyIjNiIsd2hNb2ciLy1GJTYjIiM3IyItVypRZSV5cUY0

HG solution

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

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

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

In fact there seems to be a shorter recurrence relation

n:=100: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),L70Zm),s,n/2+1)):

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

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

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

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

L01ftTm:=coeff(ser,x,2); L03ftTm:=factor(coeff(ser,x,4)); L05ftTm:=factor(coeff(ser,x,6));

LCQqKCIjOyIiIiwmSSNUbUc2IkYlRiUhIiJGJSksJkYnRiUiIiRGJUYsRilGJQ==

LCQqLCIjayIiIiksJkkjVG1HNiJGJUYlISIiIiIjRiUsJkYoRiUiIihGKkYlLCZGKEYlRiVGJUYlKSwmRihGJSIiJEYlIiInRipGKg==

LCQqKiIjOyIiIiksJkkjVG1HNiJGJUYlISIiIiIkRiUsLComIiIqRiUpRigiIiVGJUYlKiYiJC8jRiUpRihGK0YlRioqJiIkNSZGJSlGKCIiI0YlRiUqJiIldzlGJUYoRiVGJSIkcChGJUYlKSwmRihGJUYrRiVGLkYqRiU=

L01ftTm := (16*(Tm-1))/(Tm+3)^3;

LCQqKCIjOyIiIiwmSSNUbUc2IkYlRiUhIiJGJSksJkYnRiUiIiRGJUYsRilGJQ==

algL01ft:=factor(subs(t=sqrt(s),factor(resultant(numer(L01ft-L01ftTm),algTm,Tm)))):

algeqtodiffeq(algL01ft,L01ft(s));

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

de:=%:

rec:=diffeqtorec(de,L01ft(s),co(k)):

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

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{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(Tm=Tmser(n),t=sqrt(s),L01ftTm),s,n/2+1)):

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

PCUsKComLCwqJiIlL0IiIiIpSSJrRzYiIiImRihGKComIiY/OiJGKClGKiIiJUYoRigqJiImVyo9RigpRioiIiRGKEYoKiYiJms3IkYoKUYqIiIjRihGKComIiVbP0YoRipGKEYoRigtSSNjb0dGKzYjRipGKEYoKiYsLiomIiQpR0YoRilGKCEiIiomIiVnQEYoRi9GKEZCKiYiJUdoRihGM0YoRkIqJiIlTyIpRihGN0YoRkIqJiIlP1xGKEYqRihGQiIlMzVGQkYoLUY8NiMsJkYqRihGKEYoRihGKComLC4qJiIiKkYoRilGKEYoKiYiIyEqRihGL0YoRigqJiIkVyRGKEYzRihGKComIiRBJ0YoRjdGKEYoKiYiJEYmRihGKkYoRigiJG8iRihGKC1GPDYjLCZGKkYoRjhGKEYoRigvLUY8NiMiIiFGXG8vLUY8NiNGKEYo

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

LCgqLiIkYyMiIiJJImtHNiJGJSwmKiYiIiRGJUYmRiVGJSIiI0YlRiUsJkYpRiVGJUYlRiUpLCZGJkYlRitGJUYrRiUtSSNjb0dGJzYjRiZGJUYlKioiIilGJSwmKiZGK0YlRiZGJUYlRipGJUYlLCwqJiIjPUYlKUYmIiIlRiVGJSomIiQzIkYlKUYmRipGJUYlKiYiJEAjRiUpRiZGK0YlRiUqJiIkeCJGJUYmRiVGJSIjVUYlRiUtRjA2IywmRiZGJUYlRiVGJSEiIiosLCZGKUYlIiIoRiVGJSwmRiZGJUYqRiVGJSwmRilGJUYzRiVGJSlGRkYrRiUtRjA2I0YuRiVGJQ==

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

IiIh

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

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L03ftTm := -64*(Tm-1)^2*(Tm-7)*(Tm+1)/(Tm+3)^6;

LCQqLCIjayIiIiksJkkjVG1HNiJGJUYlISIiIiIjRiUsJkYoRiUiIihGKkYlLCZGKEYlRiVGJUYlKSwmRihGJSIiJEYlIiInRipGKg==

algL03ft:=factor(subs(t=sqrt(s),factor(resultant(numer(L03ft-L03ftTm),algTm,Tm)))):

algeqtodiffeq(algL03ft,L03ft(s));

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

de:=%:

rec:=diffeqtorec(de,L03ft(s),co(k)):

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

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

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMiIiQvLUYlNiNGMSIjUy8tRiU2IyIiJSIkJVwvLUYlNiMiIiYiJW9oLy1GJTYjIiInIiZkKHk=

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(Tm=Tmser(n),t=sqrt(s),L03ftTm),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 (-1,5) : no</Text-field>

L05ftTm := 16*(Tm-1)^3*(9*Tm^4-204*Tm^3+510*Tm^2+1476*Tm+769)/(Tm+3)^9;

LCQqKiIjOyIiIiksJkkjVG1HNiJGJUYlISIiIiIkRiUsLComIiIqRiUpRigiIiVGJUYlKiYiJC8jRiUpRihGK0YlRioqJiIkNSZGJSlGKCIiI0YlRiUqJiIldzlGJUYoRiVGJSIkcChGJUYlKSwmRihGJUYrRiVGLkYqRiU=

algL05ft:=factor(subs(t=sqrt(s),factor(resultant(numer(L05ft-L05ftTm),algTm,Tm)))):

algeqtodiffeq(algL05ft,L05ft(s));

LDYqJiIvU2llYTAmNCIiIiIpSSJzRzYiIiIpRiVGJSomIi4vKFwqPih6Z0YlKUYnIiIoRiUhIiIqJiIuKTM8U2YzNUYlKUYnIiInRiVGJSomIiwzUU1bcChGJSlGJyIiJkYlRi4qJiIrIVshR2NMRiUpRiciIiVGJUYlKiYiKWcwZHRGJSlGJyIiJEYlRi4qJiw2KiYiL19ScCgqZiJ5JUYlKUYnIiIqRiVGJSomIi83bFYhKXk2ISlGJUYmRiVGLiomIi8pZSVITlw5NUYlRixGJUYlKiYiLj1ZQjpSYyJGJUYxRiVGJSomIi1jOGtoKCpwRiVGNUYlRi4qJiItY2doIz1YIkYlRjlGJUYlKiYiLCNcR2MjeSJGJUY9RiVGLiomIitVIWZ6RSJGJSlGJyIiI0YlRiUqJiIpV2xLW0YlRidGJUYuIidXdHZGJUYlLUkmTDA1ZnRHRig2I0YnRiVGJSomLDgqJiIxIUc4RGBmPVwjRiUpRiciIzZGJUYlKiYiMHN1REQ1LGsiRiUpRiciIzVGJUYuKiYiMDtDeGsnKTNII0YlRkNGJUYlKiYiMF91XE49ZXoiRiVGJkYlRi4qJiIvLXBULzUqKWVGJUYsRiVGJSomIi9jP2JmbVM1RiVGMUYlRi4qJiIuKyVRaD06NkYlRjVGJUYlKiYiLGM6PSM0eEYlRjlGJUYuKiYiK3VyNylbJEYlRj1GJUYlKiYiKXcnb2kqRiVGU0YlRi4qJiIod2hBIkYlRidGJUYlRiUtSSVkaWZmRyUqcHJvdGVjdGVkRzYkRlhGJ0YlRiUqJiw4KiYiMS9SX2VyTSZbJUYlKUYnIiM3RiVGJSomIjFLIVs2amZuTSJGJUZpbkYlRi4qJiIwR0hsUlUhKUclRiVGXW9GJUYlKiYiMFtbOElfNnciRiVGQ0YlRi4qJiIvRCcpWzgiXGklRiVGJkYlRiUqJiIub1dvYHQtKEYlRixGJUYuKiYiLVdaUkhZbEYlRjFGJUYlKiYiLHdgKDRLUkYlRjVGJUYuKiYiK3hBOV86RiVGOUYlRiUqJiIpY0o1UUYlRj1GJUYuKiYiJyNIWSVGJUZTRiVGJUYlLUZicDYkRmFwRidGJUYlKiYsOComIjF3NGoqeU84NyJGJSlGJyIjOEYlRiUqJiIwbzIlZTAtOUpGJUZpcEYlRi4qJiIva0tndyYzIXpGJUZpbkYlRiUqJiIvI0g5bmlybyNGJUZdb0YlRi4qJiIuJm8nMyx0SidGJUZDRiVGJSomIi0tSnVTRSkpRiVGJkYlRi4qJiIsT0AoW0d3RiVGLEYlRiUqJiIrMz4xZ1VGJUYxRiVGLiomIipkJFJtOkYlRjVGJUYlKiYiKHFiaCRGJUY5RiVGLiomIiZzMCVGJUY9RiVGJUYlLUZicDYkRl9yRidGJUYl

de:=%:

rec:=diffeqtorec(de,L05ft(s),co(k)):

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

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

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

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCIjNS8tRiU2IyIiJSIkNSMvLUYlNiMiIiYiJTxNLy1GJTYjIiInIiYhUl4vLUYlNiMiIigiJ1ovdi8tRiU2IyIiKSIpcSlSMyIvLUYlNiMiIioiKjZxN2Mi

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(Tm=Tmser(n),t=sqrt(s),L05ftTm),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 (-3,1) : no</Text-field>

L21ftZm := -(2/3)*(Zm-1)*(Zm^6-14*Zm^5+43*Zm^4-28*Zm^3-57*Zm^2-6*Zm-3)/((Zm+1)*(Zm^2+3)^3);

n:=8: normal(series(subs(Zm=Zmser(n),L21ftZm),t,n+1));

KytJInRHNiIjIiIjIiIkRiYjIiM5RiciIiUjIiREIkYnIiInLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiKQ==

algL21ft:=factor(subs(t=sqrt(s),factor(resultant(numer(L21ft-L21ftZm),algZm,Zm)))):

de0:=algeqtodiffeq(algL21ft,L21ft(s)): nops(%);

IiIj

op(2,de0);

Ly1JJkwyMWZ0RzYiNiMiIiFGJw==

de:=de0:

rec:=diffeqtorec(de,L21ft(s),co(k)):

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

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

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subs(_C[0]=2/3,%);

PC0vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiIyIiJC8tRiU2I0YuIyIjOUYvLy1GJTYjRi8jIiREIkYvLy1GJTYjIiIlIyIlIUciRi8vLUYlNiMiIiYiJWlaLy1GJTYjIiInIiZvaiYvLUYlNiMiIigiJ3dpcC8tRiU2IyIiKSMiKSlRWm0jRi8vLUYlNiMiIiojIiplb2lbJEYvLy1GJTYjIiM1IyIrJ1EpUmFZRi8=

Some HG solutions, but not ours

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

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGTkYnRj9GJ0Y5RidGQkY5RixGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

In fact there seems to be a shorter recurrence relation

n:=160: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),L21ftZm),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) = 2/3, co(2) = 14/3, co(3) = 125/3});

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGTkYnRj9GJ0Y5RidGQkY5RixGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

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

L12Zm := -(32/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+28*Zm^6-42*Zm^5+126*Zm^4-222*Zm^3-164*Zm^2-238*Zm-119)/((Zm+1)*(Zm^2+3)^6);

algL12:=factor(subs(t=sqrt(s),factor(resultant(numer(L12-L12Zm),algZm,Zm)))):

de0:=algeqtodiffeq(algL12,L12(s)):nops(%);

IiIj

op(2,de0);

Ly1JJEwxMkc2IjYjIiIhRic=

de:=de0:

rec:=diffeqtorec(de,L12(s),co(k)):

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

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

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJiIiJC8tRiU2IyIiIyIjOy8tRiU2I0YvIiRtIi8tRiU2IyIiJSMiJXZiRi8vLUYlNiNGLiImNj8jLy1GJTYjIiInIicmKj1GLy1GJTYjIiIoIiheJG9NLy1GJTYjIiIpIilwXlBYLy1GJTYjIiIqIippJT5lZy8tRiU2IyIjNSMiLEImR1FuQ0YvLy1GJTYjIiM2Ii0vIXlMQTgiLy1GJTYjIiM3Ii4oUkcob3JkIi8tRiU2IyIjOCMiLyVbTUwueWwnRi8vLUYlNiMiIzkiMG9IYFFULTok

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),L12Zm),s,n/2+1)):

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

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

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

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJiIiJC8tRiU2IyIiIyIjOy8tRiU2I0YvIiRtIg==

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

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

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

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,0) : HG (already known, from the study of walks that START at (0,0))</Text-field>

P00Zm := -(32/3)*Zm^3*(Zm^3-3*Zm^2-Zm-1)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI0siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiJGKUYwRidGMEYnLCZGKUYnRidGJ0YwKSwmKiRGLkYnRidGJkYnRiZGMEYw

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

algeqtodiffeq(algP00,P00(s));

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

de:=%:

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

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

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

PCUvLUkjY29HNiI2IyIiISMiIiIiIiQvLUYlNiNGKiMiIiVGKy8tRiU2IyIiIyMiI0pGKw==

HG solution

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

LCYqNiMiIikiI0YiIiIpSSNQaUclKnByb3RlY3RlZEcjRiciIiMhIiIpIiIkRitGJy1JJkdBTU1BRzYkRipJKF9zeXNsaWJHNiI2IyNGLEYvRi0tRjE2IyMiIiYiIidGJykiIztJImtHRjRGJy1GMTYjLCZGPkYnRitGJ0YnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RicjIiIlRi9GJ0YtLUYxNiMsJkY+RidGLEYnRi1GJyo0I0YsRiZGJ0YoRi1GN0YtRjBGJ0Y8RidGP0YnLUYxNiMsJkY+RidGOUYnRidGTEYtLUYxNiMsJkY+RicjRjpGL0YnRi1GLQ==

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),P00Zm),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) = 4/3}));

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (2,0) : no</Text-field>

P20Zm := (32/3)*(Zm+1)*(Zm^2-5)*(Zm^4-12*Zm^3+2*Zm^2+1)*Zm^3*(Zm-1)/(Zm^2+3)^6;

LCQqMCMiI0siIiQiIiIsJkkjWm1HNiJGJ0YnRidGJywmKiQpRikiIiNGJ0YnIiImISIiRicsKiokKUYpIiIlRidGJyomIiM3RicpRilGJkYnRjAqJkYuRidGLUYnRidGJ0YnRidGN0YnLCZGKUYnRidGMEYnKSwmRixGJ0YmRiciIidGMEYn

n:=10: series(subs(Zm=Zmser(n),P20Zm),t,n+1);

Ky1JInRHNiIjIiIiIiIkIiIjIiImIiIlIiNtIiInIyIlImYjRiciIiktSSJPRyUqcHJvdGVjdGVkRzYjRiYiIzU=

algP20:=factor(subs(t=sqrt(s),factor(resultant(numer(P20Zm-P20),algZm,Zm)))):

de0:=algeqtodiffeq(algP20,P20(s)): nops(de0);

IiM+

op(2,de0);

LCQqJiIwP2JaPUFfPiIiIiIpSSJzRzYiIiM2RiUhIiI=

de:=de0:

rec:=diffeqtorec(de,P20(s),co(k)):

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

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

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

subs(_C[0]=1/3,%);

PDAvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIiImLy1GJTYjRi4iI20vLUYlNiMiIiUjIiUiZiNGLi8tRiU2I0YzIiZOOSIvLUYlNiMiIiciJydwYCIvLUYlNiMiIigiKD9lNCMvLUYlNiMiIikiKUUqXCpHLy1GJTYjIiIqIio+aVYvJS8tRiU2IyIjNSMiLFRFQD1yIkYuLy1GJTYjIiM2IixZOyRIPyIpLy1GJTYjIiM3Ii5pQEwiUms2Ly1GJTYjIiM4IyIvS15WSmlVXUYu

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),P20Zm),s,n/2+1)):

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

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

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

PCcvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIiImLy1GJTYjRi4iI20vLUYlNiMiIiUjIiUiZiNGLg==

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

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

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

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,2) : no</Text-field>

P02Zm := (32/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+22*Zm^6-54*Zm^5+60*Zm^4+42*Zm^3+322*Zm^2+86*Zm+43)/((Zm+1)*(Zm^2+3)^6);

n:=10: series(subs(Zm=Zmser(n),P02Zm),t,n+1);

Ky1JInRHNiIjIiIlIiIkIiIjIiM5RiYiJGAiIiInIyIlQ2BGJyIiKS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzU=

algP02:=factor(subs(t=sqrt(s),factor(resultant(numer(P02Zm-P02),algZm,Zm)))):

de0:=algeqtodiffeq(algP02,P02(s)): nops(de0);

IiIj

op(2,de0);

Ly1JJFAwMkc2IjYjIiIhRic=

de:=de0:

rec:=diffeqtorec(de,P02(s),co(k)):

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

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

PDEvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJSIiJC8tRiU2IyIiIyIjOS8tRiU2I0YvIiRgIi8tRiU2I0YuIyIlQ2BGLy8tRiU2IyIiJiImI2VALy1GJTYjIiInIidvQEYvLUYlNiMiIigiKDEyYCQvLUYlNiMiIikiKSJwVm8lLy1GJTYjIiIqIio5UiVIai8tRiU2IyIjNSMiLDAmKlJZZyNGLy8tRiU2IyIjNiItPkFVNDE3Ly1GJTYjIiM3Ii4oKXk9KWUkcCIvLUYlNiMiIzgjIi8vJTNNMDI/KEYvLy1GJTYjRjQiMGpQbVYtI0hN

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),P02Zm),s,n/2+1)):

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

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

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

PCcvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyIiJSIiJC8tRiU2IyIiIyIjOS8tRiU2I0YvIiRgIi8tRiU2I0YuIyIlQ2BGLw==

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

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

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

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

P11ftZm := (2/3)*(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);

n:=10: series(subs(Zm=Zmser(n), P11ftZm),t,n+1);

Ky1JInRHNiIjIiIiIiIkIiIjIyIjNUYnIiIlIyIkMSJGJyIiJyMiJTU3RiciIiktSSJPRyUqcHJvdGVjdGVkRzYjRiZGKg==

algP11ft:=factor(subs(t=sqrt(s),factor(resultant(numer(P11ft-P11ftZm),algZm,Zm)))):

de0:=algeqtodiffeq(algP11ft,P11ft(s)):nops(de0);

IiIj

de:=de0:

rec:=diffeqtorec(de,P11ft(s),co(k)):

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

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

subs(_C[0]=1/3,%);

Some HG solutions, but not ours

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

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGTkYnRj9GJ0Y5RidGQkY5RixGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

In fact there seems to be a shorter recurrence relation

n:=160: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),P11ftZm),s,n/2+1)):

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

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

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

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiI0YsIiIkLy1GJTYjIiIjIyIjNUYuLy1GJTYjRi4jIiQxIkYu

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

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

LCYqOCMiIiUiIioiIiIpIiIkI0YnIiIjRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIyIiJiIiJ0YnLUYtNiMjRitGKSEiIiwmLUkjY29HRjE2I0YnRicqJkYrRictRjw2IyIiIUYnRjlGJylJI1BpR0YvRipGOSkiIztJImtHRjFGJy1GLTYjLCZGRkYnRipGJ0YnLUYtNiMsJkZGRicjIiIoRjVGJ0YnLUYtNiMsJkZGRicjRiVGKUYnRjktRi02IywmRkZGJ0YrRidGOUYnKjYjRitGJkYnRjZGJywmKiZGK0YnRjtGJ0YnKiZGTkYnRj9GJ0Y5RidGQkY5RixGOUZERidGR0YnLUYtNiMsJkZGRidGM0YnRidGU0Y5LUYtNiMsJkZGRicjRjRGKUYnRjlGOQ==

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

B01Zm:=coeff(ser,x,0); B03Zm:=coeff(ser,x,2); B05Zm:=coeff(ser,x,4);

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

B01Zm := -(64/3)*Zm^3*(Zm^3-3*Zm^2+2*Zm+2)/((Zm+1)*(Zm^2+3)^3);

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

algB01:=factor(subs(t=sqrt(s),factor(resultant(numer(B01Zm-B01),algZm,Zm)))):

algeqtodiffeq(algB01,B01(s));

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

de:=%:

rec:=diffeqtorec(de,B01(s),co(k)):

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

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

PCUvLUkjY29HNiI2IyIiISMhIiIiIiQvLUYlNiMiIiJGKS8tRiU2IyIiI0Yp

HG solution

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

LCYqNiMiIiUiI0YiIiIpSSNQaUclKnByb3RlY3RlZEcjRiciIiMhIiIpIiIkRitGJy1JJkdBTU1BRzYkRipJKF9zeXNsaWJHNiI2IyNGLEYvRi0tRjE2IyMiIiYiIidGJykiIztJImtHRjRGJy1GMTYjLCZGPkYnRitGJ0YnLUYxNiMsJkY+RicjIiIoRjtGJ0YnLUYxNiMsJkY+RicjRiVGL0YnRi0tRjE2IywmRj5GJ0YsRidGLUYnKjQjIiM1RiZGJ0YoRi1GN0YtRjBGJ0Y8RidGP0YnLUYxNiMsJkY+RidGOUYnRidGS0YtLUYxNiMsJkY+RicjRjpGL0YnRi1GLQ==

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),B01Zm),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) = -1/3}));

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

seq((4/27)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+1/2)*GAMMA(k+7/6)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+4/3)*GAMMA(k+2))/ (fact(1/2,k)*fact(7/6,k)*16^k/fact(2,k)/fact(4/3,k)),k=1..10);

NiwjIiIjIiIqRiNGI0YjRiNGI0YjRiNGI0Yj

seq((10/27)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)/(sqrt(Pi)*GAMMA(5/6)*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);

NiwjIiImIiIqRiNGI0YjRiNGI0YjRiNGI0Yj

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

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

paper version

seq(16^k/9*(2*fact(1/2,k)*fact(7/6,k)/fact(2,k)/fact(4/3,k)-5*fact(1/2,k)*fact(5/6,k)/fact(2,k)/fact(5/3,k) +3*fact(1/2,k)^2/fact(2,k)^2),k=0..10);

Ni0iIiFGIyIiIiIjOyIkQSMiJThJIiYzNSUiJzdLYyIoN1sieSIqdXZdNCIiK3RRVFs6

coeff(coeff(convert(Cser(20),polynom),x,0),y,-2);

LDQqJiIrdFFUWzoiIiIpSSJ0RzYiIiM/RiVGJSomIip1dl00IkYlKUYnIiM9RiVGJSomIig3WyJ5RiUpRiciIztGJUYlKiYiJzdLY0YlKUYnIiM5RiVGJSomIiYzNSVGJSlGJyIjN0YlRiUqJiIlOElGJSlGJyIjNUYlRiUqJiIkQSNGJSlGJyIiKUYlRiUqJkYxRiUpRiciIidGJUYlKiQpRiciIiVGJUYl

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-4) : HG</Text-field>

B01Zm:=coeff(ser,x,0); B03Zm:=coeff(ser,x,2); B05Zm:=coeff(ser,x,4);

LCQqLCMiI2siIiQiIiIpSSNabUc2IkYmRicsKiokRihGJ0YnKiZGJkYnKUYpIiIjRichIiIqJkYvRidGKUYnRidGL0YnRicsJkYpRidGJ0YnRjApLCYqJEYuRidGJ0YmRidGJkYwRjA=

B03Zm := (64/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+25*Zm^6-48*Zm^5+93*Zm^4-90*Zm^3+79*Zm^2-76*Zm-38)/((Zm+1)*(Zm^2+3)^6);

algB03:=factor(subs(t=sqrt(s),factor(resultant(numer(B03Zm-B03),algZm,Zm)))):

algeqtodiffeq(algB03,B03(s)):

de:=%:

rec:=diffeqtorec(de,B03(s),co(k)):

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

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

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

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyEiIiIiJC8tRiU2IyIiIyEiIy8tRiU2I0YvISM4Ly1GJTYjIiIlIyEkXiNGLy8tRiU2IyIiJiEkSCUvLUYlNiMiIiciJHQj

HG solution

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

LCYqOCMiIiUiJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJykiIztJImtHRjFGJy1GLTYjLCZGOEYnRipGJ0YnLUYtNiMsJkY4RicjIiIoRjVGJ0YnLCgqJiIjQEYnKUY4RitGJ0YnKiYiI0lGJ0Y4RidGJyIjVyEiIkYnKUkjUGlHRi9GKkZILUYtNiMjRitGKUZILUYtNiMsJkY4RicjRkBGKUYnRkgtRi02IywmRjhGJ0YpRidGSEYnKjYjRitGJkYnRktGJ0Y2RidGOUYnLUYtNiMsJkY4RidGM0YnRicsKComIiR4IkYnRkRGJ0YnKiYiJCZHRidGOEYnRiciJDUiRkhGJ0YsRkhGSUZIRlJGSC1GLTYjLCZGOEYnIyIiKUYpRidGSEZI

In fact there seems to be a shorter recurrence relation

n:=80: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),B03Zm),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) = -1/3}));

LCYqOCMiIiUiJEgoIiIiKSIiJCNGJyIiI0YnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIidGJykiIztJImtHRjFGJy1GLTYjLCZGOEYnRipGJ0YnLUYtNiMsJkY4RicjIiIoRjVGJ0YnLCgqJiIjQEYnKUY4RitGJ0YnKiYiI0lGJ0Y4RidGJyIjVyEiIkYnKUkjUGlHRi9GKkZILUYtNiMjRitGKUZILUYtNiMsJkY4RicjRkBGKUYnRkgtRi02IywmRjhGJ0YpRidGSEYnKjYjRitGJkYnRktGJ0Y2RidGOUYnLUYtNiMsJkY4RidGM0YnRicsKComIiR4IkYnRkRGJ0YnKiYiJCZHRidGOEYnRiciJDUiRkhGJ0YsRkhGSUZIRlJGSC1GLTYjLCZGOEYnIyIiKUYpRidGSEZI

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

seq((4/729)*sqrt(3)*GAMMA(5/6)*16^k*GAMMA(k+1/2)*GAMMA(k+7/6)*(21*k^2+30*k-44)/(sqrt(Pi)*GAMMA(2/3)*GAMMA(k+7/3)*GAMMA(k+3))/ ((21*k^2+30*k-44)*fact(1/2,k)*fact(7/6,k)*16^k/fact(2,k+1)/fact(4/3,k+1)),k=1..10);

NiwjIiIjIiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

seq((2/729)*GAMMA(2/3)*16^k*GAMMA(k+1/2)*GAMMA(k+5/6)*(177*k^2+285*k-110)/(GAMMA(5/6)*sqrt(Pi)*GAMMA(k+3)*GAMMA(k+8/3))/ ((177*k^2+285*k-110)*fact(1/2,k)*fact(5/6,k)/fact(2,k+1)/fact(5/3,k+1)*16^k), k=1..10);

NiwjIiIiIiRWI0YjRiNGI0YjRiNGI0YjRiNGIw==

seq(Qconj(2,0,2*m)/(3*16^m*m*fact(1/2,m)^2/fact(2,m)/fact(2,m+1)),m=1..10);

NiwiIiJGI0YjRiNGI0YjRiNGI0YjRiM=

paper version

seq(16^k/3^5*(2*(21*k^2+30*k-44)*fact(1/2,k)*fact(7/6,k)/fact(2,k+1)/fact(4/3,k+1) -(177*k^2+285*k-110)*fact(1/2,k)*fact(5/6,k)/fact(2,k+1)/fact(5/3,k+1) +3^5*k*fact(1/2,k)^2/fact(2,k)/fact(2,k+1)),k=0..10);

Ni0iIiFGI0YjIiIjIiNaIiRKKSImVEwiIicpXDAjIihVVzUkIikpW2xrJSIqdTtoI3A=

coeff(coeff(convert(Cser(20),polynom),x,0),y,-4);

LDIqJiIqdTtoI3AiIiIpSSJ0RzYiIiM/RiVGJSomIikpW2xrJUYlKUYnIiM9RiVGJSomIihVVzUkRiUpRiciIztGJUYlKiYiJylcMCNGJSlGJyIjOUYlRiUqJiImVEwiRiUpRiciIzdGJUYlKiYiJEopRiUpRiciIzVGJUYlKiYiI1pGJSlGJyIiKUYlRiUqJiIiI0YlKUYnIiInRiVGJQ==

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-6) : HG</Text-field>

B05Zm := -(64/3)*Zm^3*(Zm^13-17*Zm^12+65*Zm^11-113*Zm^10+456*Zm^9-928*Zm^8+1458*Zm^7-3186*Zm^6+3105*Zm^5-3969*Zm^4+4413*Zm^3-909*Zm^2+2790*Zm+930)*(Zm-1)^2/((Zm+1)*(Zm^2+3)^9);

algB05:=factor(subs(t=sqrt(s),factor(resultant(numer(B05Zm-B05),algZm,Zm)))):

de0:=algeqtodiffeq(algB05,B05(s)):nops(de0);

IiM4

de:=de0:

rec:=diffeqtorec(de,B05(s),co(k)):

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

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

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMjISIjIiIkLy1GJTYjRjMjISNERjMvLUYlNiMiIiUjISRsI0YzLy1GJTYjIiImISQlKikvLUYlNiMiIichJVAoKS8tRiU2IyIiKCEmKmYhKS8tRiU2IyIiKSMhKEVvJz5GMy8tRiU2IyIiKiEoNktqJA==

HG solution

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

LCYqOCMiIiUiJiRvPiIiIikiIiQjRiciIiNGJy1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiImIiInRicpIiM7SSJrR0YxRictRi02IywmRjhGJ0YqRidGJy1GLTYjLCZGOEYnIyIiKEY1RidGJywsKiYiJCpbRicpRjhGJUYnRicqJiIlKVIiRicpRjhGKUYnRicqJiIlTExGJylGOEYrRichIiIqJiIlQWlGJ0Y4RidGSyIlb11GJ0YnKUkjUGlHRi9GKkZLLUYtNiMjRitGKUZLLUYtNiMsJkY4RicjIiM1RilGJ0ZLLUYtNiMsJkY4RidGJUYnRktGJyo2I0YrRiZGJ0ZRRidGNkYnRjlGJy1GLTYjLCZGOEYnRjNGJ0YnLCwqJiIlVGJGJ0ZERidGJyomIiYzeSJGJ0ZHRidGJyomIiZeLSJGJ0ZKRidGSyomIiY9IVJGJ0Y4RidGSyImIVs5RidGJ0YsRktGT0ZLLUYtNiMsJkY4RicjIiM2RilGJ0ZLRllGS0ZL

In fact there seems to be a shorter recurrence relation

n:=140: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),B05Zm),s,n/2+1)):

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

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

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

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

PCUvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiMjISIjIiIk

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

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

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (0,-8) : HG</Text-field>

B07Zm;;

algB07:=factor(subs(t=sqrt(s),factor(resultant(numer(B07Zm-B07),algZm,Zm)))):

de0:=algeqtodiffeq(algB07,B07(s)):nops(de0);

IiM6

de:=de0:

rec:=diffeqtorec(de,B07(s),co(k)):

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

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

PC8vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCMhIiZGNC8tRiU2IyIiJSMhIykqRjQvLUYlNiMiIiYhJHclLy1GJTYjIiInISVJaS8tRiU2IyIiKCEmJD14Ly1GJTYjIiIpIyEoYCt3I0Y0Ly1GJTYjIiIqISkkemUwIi8tRiU2IyIjNSEqUilcYDYvLUYlNiMiIzYhKyhIWlE7Ii8tRiU2IyIjNyMhLFkmeicpKipIRjQ=

HG solution

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

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

In fact there seems to be a shorter recurrence relation

n:=140: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),B07Zm),s,n/2+1)):

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

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

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

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCMhIiZGNA==

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

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

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

B10ftTm:=coeff(ser,x,2); B30ftTm:=coeff(ser,x,4); B50ftTm:=coeff(ser,x,6);

LCQqKCwmSSNUbUc2IiIiIiIiJiEiIkYnKSwmRiVGJ0YnRikiIiNGJyksJkYlRiciIiRGJ0YvRilGKQ==

KigsKiokKUkjVG1HNiIiIiQiIiJGKSomIiNWRikpRiYiIiNGKSEiIiomIiRKIkYpRiZGKUYpIiRuIkYpRikpLCZGJkYpRilGLkYoRikpLCZGJkYpRihGKSIiJ0Yu

LCQqKCwuKiQpSSNUbUc2IiIiJiIiIkYqKiYiJDgiRiopRiciIiVGKiEiIiomIiVRPEYqKUYnIiIkRipGKiomIiUhcCNGKilGJyIiI0YqRi8qJiIlciQqRipGJ0YqRi8iJURcRi9GKiksJkYnRipGKkYvRi5GKiksJkYnRipGM0YqIiIqRi9GLw==

B10ftTm := -(Tm-5)*(Tm-1)^2/(Tm+3)^3;

LCQqKCwmSSNUbUc2IiIiIiIiJiEiIkYnKSwmRiVGJ0YnRikiIiNGJyksJkYlRiciIiRGJ0YvRilGKQ==

algB10ft:=factor(subs(t=sqrt(s),factor(resultant(numer(B10ft-B10ftTm),algTm,Tm)))):

de0:=algeqtodiffeq(algB10ft,B10ft(s)):nops(de0);

IiIn

op(2,de0);

LCQqJiIkbyIiIiIpSSJzRzYiIiIjRiUhIiI=

de:=de0:

rec:=diffeqtorec(de,B10ft(s),co(k)):

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

{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:=100: ser:=normal(series(subs(Tm=Tmser(n),t=sqrt(s),B10ftTm),s,n/2+1)):

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

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

LCgqMCIkYyMiIiJJImtHNiJGJSwmKiYiIiRGJUYmRiVGJSIiI0YlRiUsJkYmRiVGKkYlRiUsJkYpRiVGJUYlRiUsJkYmRiVGJUYlRiUtSSNjb0dGJzYjRiZGJUYlKi4iIilGJSwmKiZGK0YlRiZGJUYlRipGJUYlLCZGJkYlRitGJUYlRi5GJSwoKiYiIz1GJSlGJkYrRiVGJSomIiNhRiVGJkYlRiUiIiZGJUYlLUYwNiNGLkYlISIiKi5GJkYlLCZGKUYlIiIoRiVGJUYsRiVGNkYlLCZGKUYlRjNGJUYlLUYwNiNGNkYlRiU=

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

PCUvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGLA==

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

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

IiIh

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

B10ftTm:=coeff(ser,x,2); B30ftTm:=coeff(ser,x,4); B50ftTm:=coeff(ser,x,6);

LCQqKCwmSSNUbUc2IiIiIiIiJiEiIkYnKSwmRiVGJ0YnRikiIiNGJyksJkYlRiciIiRGJ0YvRilGKQ==

KigsKiokKUkjVG1HNiIiIiQiIiJGKSomIiNWRikpRiYiIiNGKSEiIiomIiRKIkYpRiZGKUYpIiRuIkYpRikpLCZGJkYpRilGLkYoRikpLCZGJkYpRihGKSIiJ0Yu

LCQqKCwuKiQpSSNUbUc2IiIiJiIiIkYqKiYiJDgiRiopRiciIiVGKiEiIiomIiVRPEYqKUYnIiIkRipGKiomIiUhcCNGKilGJyIiI0YqRi8qJiIlciQqRipGJ0YqRi8iJURcRi9GKiksJkYnRipGKkYvRi5GKiksJkYnRipGM0YqIiIqRi9GLw==

B30ftTm := (Tm^3-43*Tm^2+131*Tm+167)*(Tm-1)^3/(Tm+3)^6;

KigsKiokKUkjVG1HNiIiIiQiIiJGKSomIiNWRikpRiYiIiNGKSEiIiomIiRKIkYpRiZGKUYpIiRuIkYpRikpLCZGJkYpRilGLkYoRikpLCZGJkYpRihGKSIiJ0Yu

algB30ft:=factor(subs(t=sqrt(s),factor(resultant(numer(B30ft-B30ftTm),algTm,Tm)))):

de0:=algeqtodiffeq(algB30ft,B30ft(s)):nops(de0);

IiIp

op(2,de0);

LCQqJiIqd2ROJj0iIiIpSSJzRzYiIiImRiVGJQ==

de:=de0:

rec:=diffeqtorec(de,B30ft(s),co(k)):

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

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

PCkvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCIiJS8tRiU2I0Y1IiMoKS8tRiU2IyIiJiIlZzkvLUYlNiMiIiciJmlEIw==

No HG solution

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

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

IiIh

In fact there seems to be a shorter recurrence relation

n:=100: ser:=normal(series(subs(Tm=Tmser(n),t=sqrt(s),B30ftTm),s,n/2+1)):

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

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

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

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJCIiJQ==

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

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

IiIh

<Text-field style="Heading 3" layout="Heading 3">Walks ending at (5,-1) : no</Text-field>

B50ftTm := -(Tm^5-113*Tm^4+1738*Tm^3-2690*Tm^2-9371*Tm-4925)*(Tm-1)^4/(Tm+3)^9;

LCQqKCwuKiQpSSNUbUc2IiIiJiIiIkYqKiYiJDgiRiopRiciIiVGKiEiIiomIiVRPEYqKUYnIiIkRipGKiomIiUhcCNGKilGJyIiI0YqRi8qJiIlciQqRipGJ0YqRi8iJURcRi9GKiksJkYnRipGKkYvRi5GKiksJkYnRipGM0YqIiIqRi9GLw==

algB50ft:=factor(subs(t=sqrt(s),factor(resultant(numer(B50ft-B50ftTm),algTm,Tm)))):

de0:=algeqtodiffeq(algB50ft,B50ft(s)):nops(de0);

IiM1

op(2,de0);

LCQqJiIvQ2EobyJ5TjciIiIpSSJzRzYiIiIpRiUhIiI=

de:=de0:

rec:=diffeqtorec(de,B50ft(s),co(k)):

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

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

PCwvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJEYoLy1GJTYjIiIlIiM6Ly1GJTYjIiImIiRXJS8tRiU2IyIiJyIlMCMqLy1GJTYjIiIoIiddZzsvLUYlNiMiIikiKEBneiMvLUYlNiMiIioiKTNRSlg=

No HG solution

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

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

IiIh

In fact there seems to be a shorter recurrence relation

n:=100: ser:=normal(series(subs(Tm=Tmser(n),t=sqrt(s),B50ftTm),s,n/2+1)):

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

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

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

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

PCcvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiRigvLUYlNiMiIiNGKC8tRiU2IyIiJEYoLy1GJTYjIiIlIiM6

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

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

IiIh

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

B12ftZm := (4/3)*(Zm-1)*(Zm^6-14*Zm^5+31*Zm^4-4*Zm^3-21*Zm^2-6*Zm-3)/((Zm+1)*(Zm^2+3)^3);

n:=10: series(subs(Zm=Zmser(n),B12ftZm),t,n+1);

Ky1JInRHNiIjISIiIiIkIiIjIyEiJUYnIiIlIyEjPkYnIiInIyEjcUYnIiIpLUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNQ==

algB12ft:=factor(subs(t=sqrt(s),factor(resultant(numer(B12ft-B12ftZm),algZm,Zm)))):

de0:=algeqtodiffeq(algB12ft,B12ft(s)):nops(de0);

IiIj

op(2,de0);

Ly1JJkIxMmZ0RzYiNiMiIiFGJw==

de:=de0:

rec:=diffeqtorec(de,B12ft(s),co(k)):

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

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

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

No HG solution

subs(_C[0]=-1/3,%);

PC0vLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyEiIiIiJC8tRiU2IyIiIyMhIiVGLy8tRiU2I0YvIyEjPkYvLy1GJTYjIiIlIyEjcUYvLy1GJTYjIiImIiQsIi8tRiU2IyIiJyIldVkvLUYlNiMiIigiJnpPKi8tRiU2IyIiKSMiKCdSZ1pGLy8tRiU2IyIiKiMiKVkjSGMoRi8vLUYlNiMiIzUjIitXKGZrOyJGLw==

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:=160: ser:=normal(series(subs(Zm=Zmser(n),t=sqrt(s),B12ftZm),s,n/2+1)):

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

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

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

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

PCYvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyEiIiIiJC8tRiU2IyIiIyMhIiVGLy8tRiU2I0YvIyEjPkYv

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

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

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

B21Zm := (64/3)*Zm^3*(Zm-1)*(Zm^8-10*Zm^7+Zm^6+93*Zm^4-42*Zm^3+7*Zm^2-76*Zm-38)/((Zm+1)*(Zm^2+3)^6);

algB21:=factor(subs(t=sqrt(s),factor(resultant(numer(B21Zm-B21),algZm,Zm)))):

de0:=algeqtodiffeq(algB21,B21(s)):nops(de0);

IiIj

op(2,de0);

Ly1JJEIyMUc2IjYjIiIhRic=

de:=de0:

rec:=diffeqtorec(de,B21(s),co(k)):

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

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

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HG solution

(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),B21Zm),s,n/2+1)):

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

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

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

PCcvLUkjY29HNiI2IyIiIUYoLy1GJTYjIiIiIyEiIiIiJC8tRiU2IyIiI0YuLy1GJTYjRi8iIicvLUYlNiMiIiUjIiREJ0Yv

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

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