In this section we allow steps between (-1,0) and (0,-1), and back restart:

The kernel K:=1-t*(x+1/x+y+1/y+(x+1/x)*(y+1/y)): Main functional equation. Cm stands for C_minus eqC:=-K*C(x,y)+1-t/y*(x+1/x+1)*Cm(1/x)-t/x*(y+1/y+1)*Cm(1/y)-t/x/y*C00+ t*(1/x+1/y)*Cm10;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsLiomLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxKiRGNSEiIkYxSSJ5R0YoRjEqJEY4RjdGMSomLCZGNUYxRjZGMUYxLCZGOEYxRjlGMUYxRjFGMUY3RjEtSSJDR0YoNiRGNUY4RjFGN0YxRjEqKkYzRjFGOEY3LChGNUYxRjZGMUYxRjFGMS1JI0NtR0YoNiNGNkYxRjcqKkYzRjFGNUY3LChGOEYxRjlGMUYxRjFGMS1GQzYjRjlGMUY3KipGM0YxRjVGN0Y4RjdJJEMwMEdGKEYxRjcqKEYzRjEsJkY2RjFGOUYxRjFJJUNtMTBHRihGMUYxNyNGLg==

Expansions Cmex:=proc(n) local res,i: res:=coeff(convert(Cex(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n): end: C00ex:=proc(n) local res: res:=coeff(convert(Cex(n),polynom),y,0): coeff(coeff(res,x,0),y,0): end: Cex:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C00=C00ex(n-1),C(x,y)=Cex(n-1),Cm(1/x)=Cmex(n-1),Cm(1/y)=subs(x=y,Cmex(n-1)),Cm10=coeff(Cmex(n-1),x,-1),eqC+C(x,y)),t,n+1)): fi: end: coeff(Cmex(2),x,-1)

LCYqJiIiJCIiIilJInRHNiIiIiNGJUYlRidGJQ==

map(expand,Cex(2));

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

All walks in C: this sequence does not appear in the OEIS n:=15:subs(x=1,y=1,Cex(n));

K0NJInRHNiIiIiIiIiEiIihGJSIjXyIiIyIkJlEiIiQiJTlIIiIlIiZnQCMiIiYiJ0MqcCIiIiciKCZcMzhGJyIpUSY+LCIiIikiKSV5KyZ5IiIqIiovJz4xaCIjNSIrOSs7Z1oiIzYiLF82ZXpyJCIjNyItaSVlI28zSCIjOCIuUTw5Pyl5QSIjOSIvdE8qcCNmKHkiIiM6

seq(coeff(%,t,i),i=0..15);

NjIiIiIiIigiI18iJCZRIiU5SCImZ0AjIidDKnAiIigmXDM4IilRJj4sIiIpJXkrJnkiKi8nPjFoIis5KztnWiIsXzZlenIkIi1pJWUjbzNIIi5RPDk/KXlBIi90TypwI2YoeSI=

Here is what we had without the diagonal step between (-1,0) and (0,-1) 1, 7, 50, 369, 2772, 21020, 160736, 1235585, 9541756, 73937340, 574600088, 4476053040, 34938781088, 273193901254, 2139381829992, 16775549720527

NjIiIiIiIigiI10iJHAkIiVzRiImPzUjIidPMjsiKCZlTjciKGM8YSoiKVN0JFIoIiopMytZZCIrU0kwd1ciLCkzInlRXCQiLWE3IVI+dCMiLiMqKkg9UVJAIi9GMHNcYng7

Walks ending at (0,0). This one is not in the OEIS either n:=30:coeff(coeff(convert(Cex(n),polynom),x,0),y,0);

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

seq(coeff(%,t,i),i=0..15);

NjIiIiIiIiEiIigiIz8iJGsiIiRpKSIlSGYiJnd0JCInaVtEIigpbz48IilJciQ+IiIpRUtHJCkiKkVQLSplIitHaEEmPiUiLGIhW0g4SSItbTtSJXk8Iw==

Walks in the quadrant. Qp(x) stands for Q(x,0) eqQ:=-K*Q(x,y)+1-t/y*(x+1/x+1)*Qp(x)-t/x*(y+1/y+1)*Qp(y)+t/x/y*Q00;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxKiRGNSEiIkYxSSJ5R0YoRjEqJEY4RjdGMSomLCZGNUYxRjZGMUYxLCZGOEYxRjlGMUYxRjFGMUY3RjEtSSJRR0YoNiRGNUY4RjFGN0YxRjEqKkYzRjFGOEY3LChGNUYxRjZGMUYxRjFGMS1JI1FwR0YoNiNGNUYxRjcqKkYzRjFGNUY3LChGOEYxRjlGMUYxRjFGMS1GQzYjRjhGMUY3KipGM0YxRjVGN0Y4RjdJJFEwMEdGKEYxRjE3I0Yu

Expansions Qpex:=proc(n) subs(y=0,Qex(n)): end: Q00ex:=proc(n) subs(x=0,y=0,Qex(n)): end: Qex:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q00=Q00ex(n-1),Q(x,y)=Qex(n-1),Qp(x)=Qpex(n-1),Qp(y)=subs(x=y,Qpex(n-1)),eqQ+Q(x,y)),t,n+1)): fi: end: Qex(4);

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

eqorbitQ:=KK*(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);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcW9yYml0UUdGKCwmKiZJI0tLR0YoIiIiLCoqKEkieEdGKEYxSSJ5R0YoRjEtSSJRR0YoNiRGNEY1RjFGMSooRjVGMUY0ISIiLUY3NiQqJEY0RjpGNUYxRjoqKEY0RjpGNUY6LUY3NiRGPSokRjVGOkYxRjEqKEY0RjFGNUY6LUY3NiRGNEZBRjFGOkYxRjEqJiwmRjRGMUY9RjpGMSwmRjVGMUZBRjpGMUY6NyNGLg==

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

An arbitrary linear combination, with all 4 solutions Cexpr:=A(x,y)+a1*Q(x,y)-a2/x^2*Q(1/x,y)+a3/x^2/y^2*Q(1/x,1/y)-a4/y^2*Q(x,1/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZDZXhwckdGKCwsLUkiQUdGKDYkSSJ4R0YoSSJ5R0YoIiIiKiZJI2ExR0YoRjQtSSJRR0YoRjFGNEY0KihJI2EyR0YoRjQpRjIiIiMhIiItRjg2JCokRjJGPUYzRjRGPSoqSSNhM0dGKEY0RjtGPSlGM0Y8Rj0tRjg2JEZAKiRGM0Y9RjRGNCooSSNhNEdGKEY0RkNGPS1GODYkRjJGRkY0Rj03I0Yu

Cmxexpr:=Am(1/x)-a2/x^2*Qp(1/x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShDbXhleHByR0YoLCYtSSNBbUdGKDYjKiRJInhHRighIiIiIiIqKEkjYTJHRihGNSlGMyIiI0Y0LUkjUXBHRihGMUY1RjQ3I0Yu

Cmyexpr:=Am(1/y)-a4*Qp(1/y)/y^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShDbXlleHByR0YoLCYtSSNBbUdGKDYjKiRJInlHRighIiIiIiIqKEkjYTRHRihGNS1JI1FwR0YoRjFGNSlGMyIiI0Y0RjQ3I0Yu

eqC;eqQ;

LC4qJiwmIiIiRiUqJkkidEc2IkYlLCxJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiUqJiwmRipGJUYrRiVGJSwmRi1GJUYuRiVGJUYlRiVGLEYlLUkiQ0dGKDYkRipGLUYlRixGJUYlKipGJ0YlRi1GLCwoRipGJUYrRiVGJUYlRiUtSSNDbUdGKDYjRitGJUYsKipGJ0YlRipGLCwoRi1GJUYuRiVGJUYlRiUtRjg2I0YuRiVGLCoqRidGJUYqRixGLUYsSSRDMDBHRihGJUYsKihGJ0YlLCZGK0YlRi5GJUYlSSVDbTEwR0YoRiVGJQ==

LCwqJiwmIiIiRiUqJkkidEc2IkYlLCxJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiUqJiwmRipGJUYrRiVGJSwmRi1GJUYuRiVGJUYlRiVGLEYlLUkiUUdGKDYkRipGLUYlRixGJUYlKipGJ0YlRi1GLCwoRipGJUYrRiVGJUYlRiUtSSNRcEdGKDYjRipGJUYsKipGJ0YlRipGLCwoRi1GJUYuRiVGJUYlRiUtRjg2I0YtRiVGLCoqRidGJUYqRixGLUYsSSRRMDBHRihGJUYl

Forming an equation for A(x,y). We want the a_i to sum to 1 to have a vanishing orbit sum for xyA(x,y) eqA:=collect(subs( C00=A00+a1*Q00,Cm10=Am10,C(x,y)=Cexpr,Cm(1/x)=Cmxexpr,Cm(1/y)=Cmyexpr,a1=1-a2-a3-a4, eqC-a1*eqQ+a2/x^2*subs(x=1/x,eqQ)+a4/y^2*subs(y=1/y,eqQ)-a3/x^2/y^2*subs(x=1/x,y=1/y,eqQ)),[A,Am,Q,Qp,Q00,A00,Am10],factor);

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

We want this not to involve Q. Consider the coefficient of Qp(1/x) eqA:=subs(a3=0,eqA);

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

Then of Qp(x) eqA:=map(factor,subs(a2=1-2*a4,eqA));

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

Finally of Qp(y) eqA:=map(factor,subs(a4=1/3,eqA));

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

A linear combination Aex:=proc(n) expand(series(Cex(n)-1/3*(Qex(n)-1/x^2*subs(x=1/x,Qex(n))-1/y^2*subs(y=1/y,Qex(n))),t,n+1)): end: Aex(2);

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

Checking that A has orbit sum zero n:=6: factor(series(x*y*Aex(n)-y/x*subs(x=1/x,Aex(n))-x/y*subs(y=1/y,Aex(n))+1/x/y*subs(x=1/x,y=1/y,Aex(n)),t,n+1));

IiIh

Cexpr:=subs(a2=a1,a4=a1,a3=0,a1=1/3,Cexpr);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZDZXhwckdGKCwqLUkiQUdGKDYkSSJ4R0YoSSJ5R0YoIiIiKiYjRjQiIiRGNC1JIlFHRihGMUY0RjQqKEY2RjQpRjIiIiMhIiItRjk2JCokRjJGPUYzRjRGPSooRjZGNClGM0Y8Rj0tRjk2JEYyKiRGM0Y9RjRGPTcjRi4=

Cmxexpr:=Am(1/x)-1/3/x^2*Qp(1/x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShDbXhleHByR0YoLCYtSSNBbUdGKDYjKiRJInhHRighIiIiIiIqKCNGNSIiJEY1KUYzIiIjRjQtSSNRcEdGKEYxRjVGNDcjRi4=

Cmyexpr:=Am(1/y)-1/3*Qp(1/y)/y^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShDbXlleHByR0YoLCYtSSNBbUdGKDYjKiRJInlHRighIiIiIiIqKCNGNSIiJEY1LUkjUXBHRihGMUY1KUYzIiIjRjRGNDcjRi4=

eqA;

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

Am is negative in x Amex:=proc(n) local res,i: res:=coeff(convert(Aex(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: A00ex:=proc(n) coeff(coeff(convert(Aex(n),polynom),y,0),x,0):end: Am10ex:=proc(n) coeff(coeff(convert(Aex(n),polynom),y,0),x,-1):end: n:=16: normal(series(subs(A00=A00ex(n),Am10=Am10ex(n),A(x,y)=Aex(n), Am(1/x)=Amex(n), Am(1/y)=subs(x=y,Amex(n)), eqA),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzw=

expand(3*A00ex(10));

LDYqJiIpS215TSIiIilJInRHNiIiIzVGJUYlKiYiKHM/KlxGJSlGJyIiKkYlRiUqJiIneWl0RiUpRiciIilGJUYlKiYiJylIMiJGJSlGJyIiKEYlRiUqJiImIylvIkYlKUYnIiInRiVGJSomIiVFQ0YlKUYnIiImRiVGJSomIiRhJUYlKUYnIiIlRiVGJSomIiNhRiUpRiciIiRGJUYlKiYiIz1GJSlGJyIiI0YlRiVGSUYl

What we need to remember from the above sub-section: Aex:=proc(n) expand(series(Cex(n)-1/3*(Qex(n)-1/x^2*subs(x=1/x,Qex(n))-1/y^2*subs(y=1/y,Qex(n))),t,n+1)): end: eqA:=(t*x^2*y^2+t*x^2*y+t*x*y^2+t*x^2+t*y^2+t*x+t*y-x*y+t)*A(x, y)/(x*y)-t*(x^2+x+1)*Am(1/x)/(x*y)-t*(y^2+y+1)*Am(1/y)/(x*y)-t*A00/(x*y)+(2*x^2*y^2+x^2+y^2)/(3*x^2*y^2)+(t *(y+x)* Am10)/(x *y);

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

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

LCgqJiw6KiYiIiMiIiIpSSJ5RzYiRiZGJ0YnKiZGJkYnRilGJ0YnIiIkRicqKCMiIiZGLEYnSSJ4R0YqRidGKEYnRicqKCMiIilGLEYnRjBGJ0YpRidGJyomRixGJ0YwRidGJyooI0YmRixGJylGMEYmRidGKUYnRicqKEY2RidGN0YnRihGJ0YnKiZGNkYnRjdGJ0YnKigjRidGLEYnKUYwRixGJ0YoRidGJyooRjZGJ0Y8RidGKUYnRicqJkY2RidGPEYnRidGJylJInRHRipGJkYnRicqJiwsRilGJ0YnRicqKEY7RidGMEYnRilGJ0YnKiZGO0YnRjdGJ0YnKihGO0YnRjdGJ0YpRidGJ0YnRkBGJ0YnKiZGO0YnRjBGJ0Yn

P(x,y) is the non-negative part of A in x and y Pex:=proc(n) local res,i,j: res:=convert(Aex(n),polynom): collect(add(add(coeff(coeff(res,x,i),y,j)*x^i*y^j,i=0..n),j=0..n),t): end: Pex(3);

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

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);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcW9yYml0QUdGKCwqKihJInhHRigiIiJJInlHRihGMS1JIkFHRig2JEYwRjJGMUYxKihGMkYxRjAhIiItRjQ2JCokRjBGN0YyRjFGNyooRjBGN0YyRjctRjQ2JEY6KiRGMkY3RjFGMSooRjBGMUYyRjctRjQ2JEYwRj5GMUY3NyNGLg==

Expression of A in terms of M and P AexprPM:=A(x,y)=P(x,y)+M(1/x,y)/x+M(1/y,x)/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShBZXhwclBNR0YoLy1JIkFHRig2JEkieEdGKEkieUdGKCwoLUkiUEdGKEYxIiIiKiYtSSJNR0YoNiQqJEYyISIiRjNGN0YyRj1GNyomLUY6NiQqJEYzRj1GMkY3RjNGPUY3NyNGLg==

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

LDoqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYtSSJNR0YlNiQqJEYkISIiRidGJkYmKiZGJEYmLUYtNiQqJEYnRjBGJEYmRiYqKEYkRjBGJ0YmLUYpRi5GJkYwKiZGJ0YmLUYtRipGJkYwKiZGJEYwLUYtNiRGNEYvRiZGMCooRiRGMEYnRjAtRik2JEYvRjRGJkYmKiZGJ0YwLUYtNiRGJEY0RiZGJiomRiRGMC1GLTYkRidGL0YmRiYqKEYkRiZGJ0YwLUYpRkFGJkYwKiZGJ0YwLUYtRj5GJkYwKiZGJEYmLUYtNiRGJ0YkRiZGMA==

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));

LCgqKEkieEc2IiIiIkkieUdGJUYmLUkiUEdGJTYkRiRGJ0YmRiYqJkYnRiYsJi1JIk1HRiVGKkYmLUYuNiQiIiFGJyEiIkYmRjIqJkYkRiYsJi1GLjYkRidGJEYmLUYuNiRGMUYkRjJGJkYy

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdQZXhwck1HRigsKiomSSJ4R0YoISIiLUkiTUdGKDYkRjBJInlHRigiIiJGNiomRjBGMS1GMzYkIiIhRjVGNkYxKiZGNUYxLUYzNiRGNUYwRjZGNiomRjVGMS1GMzYkRjpGMEY2RjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdBZXhwck1HRigvLUkiQUdGKDYkSSJ4R0YoSSJ5R0YoLC4qJkYyISIiLUkiTUdGKEYxIiIiRjkqJkYyRjYtRjg2JCIiIUYzRjlGNiomRjNGNi1GODYkRjNGMkY5RjkqJkYzRjYtRjg2JEY9RjJGOUY2KiYtRjg2JCokRjJGNkYzRjlGMkY2RjkqJi1GODYkKiRGM0Y2RjJGOUYzRjZGOTcjRi4=

Checking the equivalence with the orbit equation for A expand(subs(AexprM,subs(x=1/x,AexprM),subs(y=1/y,x=1/x,AexprM),subs(y=1/y,AexprM),eqorbitA));

IiIh

Let us check the expression of P(x,y) in terms of M PexprM;

LCoqJkkieEc2IiEiIi1JIk1HRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1GKDYkRipGJEYrRisqJkYqRiYtRig2JEYvRiRGK0Ym

n:=4: collect(Mex(n)/x+subs({x=y,y=x},Mex(n))/y-1/x*subs(x=0,Mex(n))-1/y*subs(x=0,y=x,Mex(n)),t,normal);

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

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

IiIh

We write an equation for P(x,y), using its combinatorial interpretation in terms of walks (unchanged) eqP:=-K*P(x,y)+2/3-t/y*(x+1/x+1)*P(x,0)-t/x*(1+y+1/y)*P(0,y)+t/x/y*P00 +t*(x+1/x+1)*M(0,x)-t/x*M00 + t*(1+y+1/y)*M(0,y)-t/y*M00;

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

n:=10: normal(series(subs(P(x,y)=Pex(n), P(x,0)=subs(y=0,Pex(n)), P(0,y)=subs(x=0,Pex(n)),P00=subs(x=0,y=0,Pex(n)),M(0,y)=subs(x=0,Mex(n)), M00=subs(x=0,y=0,Mex(n)), M(0,x)=subs(x=0,y=x,Mex(n)),eqP),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

An equation for M (changed), one new term eqM1:=-K/x*M(1/x,y)+1/3/x^2-t*(1+1/x+x)/y/x*M(1/x,0)-t*(y+1/y+1)*M(0,y)+t/y*M(0,0)+t*(1/y+1+y)/x*P(0,y)-t/x/y*P(0,0)+ t/x*M(0,0);

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

eqM2:=subs(x=1/x,eqM1);

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

Let us check this second equation n:=10: normal(series(subs(M(x,y)=Mex(n),M(0,y)=subs(x=0,Mex(n)),M(x,0)=subs(y=0,Mex(n)), M(0,0)=subs(x=0,y=0,Mex(n)),P(0,y)=subs(x=0,Pex(n)), P(0,0)=subs(x=0,y=0,Pex(n)), eqM2),t, n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

One more ingredient PexprM;

LCoqJkkieEc2IiEiIi1JIk1HRiU2JEYkSSJ5R0YlIiIiRisqJkYkRiYtRig2JCIiIUYqRitGJiomRipGJi1GKDYkRipGJEYrRisqJkYqRiYtRig2JEYvRiRGK0Ym

limit(%,x=0);

KiYsKComLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiI2I0kiTUdGLDYkIiIhSSJ5R0YsRi5GM0YuRi4tRjA2JEYzRjJGLi1GMDYkRjJGMiEiIkYuRjNGOA==

limit(%,y=0);

LCQqJiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiU2I0kiTUdGLTYkIiIhRjJGJUYl

eqM3:=collect(subs(P(0,y)=%%,P(0,0)=%,eqM2/x),[P,M],factor);

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

We now expand the equation for M around x=0 normal(series(eqM3,x,2));

KydJInhHNiIsJComLDoqKiIiIyIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRio2I0kiTUdGJDYkIiIhSSJ5R0YkRipJInRHRiRGKilGNyIiJEYqISIiKigtRjRGNUYqRjhGKkY5RipGOyoqRilGKkYrRipGOEYqKUY3RilGKkY7KigtRjQ2JEY3RjZGKkY4RipGP0YqRjsqKkY6RipGOEYqLUYsNiRGNkY2RipGN0YqRioqKkYpRipGOEYqRitGKkY3RipGOyooRkFGKkY4RipGN0YqRjsqKkYpRipGOEYqLUY0RkVGKkY3RipGKiooRj1GKkY4RipGN0YqRjsqJkY9RipGP0YqRioqJkY4RipGQUYqRjsqJkY4RipGSUYqRipGKkY/RjtGO0Y2LUkiT0dGL0YyRio=

This gives an expression for M_x(0,y) collect(coeff(%,x,0),[M,(D[1](M))(0, y)],factor);

LCwqKkkidEc2IiIiIiwmKiYiIiNGJkkieUdGJUYmRiZGJkYmRiYpRipGKSEiIi1JIk1HRiU2JCIiIUYwRiZGLCooLCgqJkYkRiZGK0YmRiZGJEYmRipGLEYmRipGLC1GLjYkRjBGKkYmRiYqKkYkRiYsKCokRitGJkYmRipGJkYmRiZGJkYrRiwtRi42JEYqRjBGJkYmKioiIiRGJkYkRiZGKkYsLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJjYjRi5GL0YmRiwqLEYpRiZGJEYmRjdGJkYqRiwtRj5GNUYmRiY=

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

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIiNiNJIk1HRio2JCIiIUkieUdGKiwkKiwjRiwiIiNGLCwqKipJInRHRipGLCwmKiZGNUYsRjFGLEYsRixGLEYsKUYxRjUhIiItRi42JEYwRjBGLEYsKigsKComRjhGLEY7RixGLEY4RixGMUY8RixGMUY8LUYuRi9GLEY8KipGOEYsLCgqJEY7RixGLEYxRixGLEYsRixGO0Y8LUYuNiRGMUYwRixGPCoqIiIkRixGOEYsRjFGPC1GJEY+RixGLEYsRjhGPEZERjxGMUYsRiw=

eqM:=collect(subs(%,eqM3),[M],factor);

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

What we need to remember eqM;

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

Introduce a series N to obtain slightly smaller coefficients eqN:=collect(subs(M(x,y)=N(x,y)+1/2*M(0,y),eqM),[M,N],factor);

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

The roots of the kernel Yens:=factor([solve(K,y)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZZW5zR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwkKiYsKkY1RjIqKEYzRjJGNkYyRjhGMkYyRjZGMkY4RjJGMiwqKigiIiRGMkY2RjJGN0YyRjJGP0YyKiZGQkYyRjZGMkYyRjhGOUYyRjlGMUYyRjJGMkY2RjksKCokRjdGMkYyRjhGMkYyRjJGOUYyLCQqKkYxRjIsKkY1RjJGOkYyRjZGMkY4RjlGMkY2RjlGREY5Rjk3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZZMHNvbEdGKCwkKiojIiIiIiIjRjEsKiomSSJ0R0YoRjEpSSJ4R0YoRjJGMUYxKiQpLCQqJiwqRjRGMSooRjJGMUY1RjFGN0YxRjFGNUYxRjdGMUYxLCoqKCIiJEYxRjVGMUY2RjFGMUY9RjEqJkZARjFGNUYxRjFGNyEiIkYxRkJGMEYxRjFGNUYxRjdGQkYxRjVGQiwoKiRGNkYxRjFGN0YxRjFGMUZCRkI3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZZMXNvbEdGKCwkKiojIiIiIiIjRjEsKiomSSJ0R0YoRjEpSSJ4R0YoRjJGMSEiIkY1RjhGN0YxKiQpLCQqJiwqRjRGMSooRjJGMUY1RjFGN0YxRjFGNUYxRjdGMUYxLCoqKCIiJEYxRjVGMUY2RjFGMUY+RjEqJkZBRjFGNUYxRjFGN0Y4RjFGOEYwRjFGMUYxRjVGOCwoKiRGNkYxRjFGN0YxRjFGMUY4RjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwmKiQpLCYiIiJGMiomSSJ0R0YoRjIsJkkieEdGKEYyKiRGNiEiIkYyRjJGOCIiI0YyRjIqKCIiJUYyKUY0RjlGMiksKEY2RjJGN0YyRjJGMkY5RjJGODcjRi4=

Check the expression of the paper Ypap:=(1-t*(x+1/x)-sqrt(Delta))/2/t/(1/x+x+1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZcGFwR0YoLCQqKiMiIiIiIiNGMSwoRjFGMSomSSJ0R0YoRjEsJkkieEdGKEYxKiRGNyEiIkYxRjFGOSokKSwmKiQpLCZGMUYxRjRGOUYyRjFGMSooIiIlRjEpRjVGMkYxKSwoRjdGMUY4RjFGMUYxRjJGMUY5RjBGMUY5RjFGNUY5RkRGOUYxNyNGLg==

simplify(Ypap/Y0sol,symbolic);

IiIi

Y0ser:=proc(n) subs(csgn(x)=1,simplify(series(Y0sol,t,n+2),symbolic)): end: Y0ser(4);

Ky1JInRHNiIqJiwoKiQpSSJ4R0YkIiIjIiIiRitGKUYrRitGK0YrRikhIiJGKyooRiZGKywmRidGK0YrRitGK0YoRixGKiomLDAqJkYqRispRikiIidGK0YrKiYiIiVGKylGKSIiJkYrRisqJiIiKkYrKUYpRjVGK0YrKiZGOUYrKUYpIiIkRitGKyomRjlGK0YoRitGKyomRjVGK0YpRitGK0YqRitGK0Y8RixGPSooLDAqJkY1RitGMkYrRisqJiIjNUYrRjZGK0YrKiYiI0BGK0Y6RitGKyomIiNCRitGPEYrRisqJkZGRitGKEYrRisqJkZERitGKUYrRitGNUYrRitGLkYrRjpGLEY1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGNw==

factor(expand(Y0sol*Y1sol));

IiIi

The following pairs cancel the kernel, and can be substituted for (x,y) in the equation for M(x,y) : (x,Y), (1/x,Y), (Y,x) and (Y, 1/x) f1:=collect(subs(N(x,y)=0,y=Y,eqN),[M]); f2:=subs(x=1/x,f1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMUdGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwmSSJ4R0YoRjEqJkYyRjFJIllHRihGMSEiIkYxRjVGOClGN0YyRjgtSSJNR0YoNiQiIiFGPUYxRjgqMkYwRjFGM0YxLCgqJEY5RjFGMUY3RjFGMUYxRjEsJkY1RjFGMUY4RjEsJkY1RjFGMUYxRjFGNUY4RjdGOC1GOzYkRj1GN0YxRjEqLEYzRjEsKCokKUY1RjJGMUYxRjVGMUYxRjFGMUY1RjhGN0Y4LUY7NiRGNUY9RjFGOCosRjBGMUYzRjFGP0YxRjlGOC1GOzYkRjdGPUYxRjEqKkYwRjFGM0YxRjdGOC0tJkkiREdGJTYjRjE2I0Y7RjxGMUY4KiYjRjEiIiRGMUY1RjFGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMkdGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwmKiRJInhHRighIiJGMSomRjJGMUkiWUdGKEYxRjdGMUY2RjEpRjlGMkY3LUkiTUdGKDYkIiIhRj5GMUY3KjJGMEYxRjNGMSwoKiRGOkYxRjFGOUYxRjFGMUYxLCZGNUYxRjFGN0YxLCZGNUYxRjFGMUYxRjZGMUY5RjctRjw2JEY+RjlGMUYxKixGM0YxLCgqJClGNkYyRjdGMUY1RjFGMUYxRjFGNkYxRjlGNy1GPDYkRjVGPkYxRjcqLEYwRjFGM0YxRkBGMUY6RjctRjw2JEY5Rj5GMUYxKipGMEYxRjNGMUY5RjctLSZJIkRHRiU2I0YxNiNGPEY9RjFGNyomI0YxIiIkRjFGNkY3RjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmM0dGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwmSSJZR0YoRjEqJkYyRjFJInhHRihGMSEiIkYxRjVGOClGN0YyRjgtSSJNR0YoNiQiIiFGPUYxRjgqMkYwRjFGM0YxLCgqJEY5RjFGMUY3RjFGMUYxRjEsJkY1RjFGMUY4RjEsJkY1RjFGMUYxRjFGNUY4RjdGOC1GOzYkRj1GN0YxRjEqLEYzRjEsKCokKUY1RjJGMUYxRjVGMUYxRjFGMUY1RjhGN0Y4LUY7NiRGNUY9RjFGOCosRjBGMUYzRjFGP0YxRjlGOC1GOzYkRjdGPUYxRjEqKkYwRjFGM0YxRjdGOC0tJkkiREdGJTYjRjE2I0Y7RjxGMUY4KiYjRjEiIiRGMUY1RjFGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmNEdGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwmSSJZR0YoRjEqJkYyRjFJInhHRighIiJGOEYxRjVGOClGN0YyRjEtSSJNR0YoNiQiIiFGPUYxRjgqMkYwRjFGM0YxLCgqJEY5RjhGMSokRjdGOEYxRjFGMUYxLCZGNUYxRjFGOEYxLCZGNUYxRjFGMUYxRjVGOEY3RjEtRjs2JEY9RkFGMUYxKixGM0YxLCgqJClGNUYyRjFGMUY1RjFGMUYxRjFGNUY4RjdGMS1GOzYkRjVGPUYxRjgqLEYwRjFGM0YxRj9GMUY5RjEtRjs2JEZBRj1GMUYxKipGMEYxRjNGMUY3RjEtLSZJIkRHRiU2I0YxNiNGO0Y8RjFGOComI0YxIiIkRjFGNUYxRjE3I0Yu

indets(f1*f2*f3*f4);

PCtJIllHNiJJInRHRiRJInhHRiQtSSJNR0YkNiQiIiFGIy1GKDYkRipGJi1GKDYkRioqJEYmISIiLUYoNiRGI0YqLUYoNiRGJkYqLUYoNiRGL0Yq

We first eliminate the two series involving Y MYsol:=factor(solve({f1,f2},{M(0,Y),M(Y,0)}));

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

Here are our two remaining equations f3bis:=collect(1/3/t/x/Y*numer(factor(subs(MYsol,f3))),[M],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZmM2Jpc0dGKCwuKigsJiomIiIjIiIiKUkieEdGKEYyRjNGM0YzRjNGM0Y1ISIiLUkiTUdGKDYkIiIhRjpGM0YzKiwsKCokRjRGM0YzRjVGM0YzRjNGMywmSSJZR0YoRjNGM0Y2RjMsJkY/RjNGM0YzRjNGP0Y2LUY4NiRGOkY1RjNGMyooRjxGM0Y1RjYtRjg2JEY1RjpGM0Y2KipGMkYzRjxGM0Y1RjYtRjg2JCokRjVGNkY6RjNGNiomIiIkRjMtLSZJIkRHRiU2I0YzNiNGOEY5RjNGNiosI0YyRktGM0Y/RjNGMEYzSSJ0R0YoRjZGNUY2RjM3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZmNGJpc0dGKCwuKiYsJiokKUkieEdGKCIiIyIiIkY1RjRGNUY1LUkiTUdGKDYkIiIhRjlGNUY1Ki4sKEYxRjVGM0Y1RjVGNUY1LCZJIllHRihGNUY1ISIiRjUsJkY9RjVGNUY1RjVGPUY+RjNGPi1GNzYkRjkqJEYzRj5GNUY1KiYsKComRjRGNUYyRjVGPiomRjRGNUYzRjVGPkY0Rj5GNS1GNzYkRjNGOUY1RjUqJiwoRjFGPkYzRj5GNUY+RjUtRjc2JEZCRjlGNUY1KigiIiRGNUYzRjUtLSZJIkRHRiU2I0Y1NiNGN0Y4RjVGPioqI0Y0Rk5GNUY9RjVGMEY1SSJ0R0YoRj5GNTcjRi4=

They only differ by x->1/x factor(subs(x=1/x,f3bis)/f4bis);

KiRJInhHNiIhIiI=

Let us check n:=10: normal(series(subs(M(0,x)=subs(x=0,y=x,Mex(n)),M(0,1/x)=subs(x=0,y=1/x,Mex(n)), M(x,0)=subs(y=0,Mex(n)), M(1/x,0)=subs(x=1/x,y=0,Mex(n)), (D[1](M))(0, 0)=coeff(subs(y=0,Mex(n)),x,1),M(0,0)=coeff(subs(y=0,Mex(n)),x,0),Y=Y0ser(n+2),f3bis),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

Eliminate M(1/x,0) f8:=collect(-f3bis+2/x*f4bis,[M],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmOEdGKCwuKigiIiQiIiJJInhHRighIiItSSJNR0YoNiQiIiFGN0YxRjEqLCwoKiQpRjIiIiNGMUYxRjJGMUYxRjFGMSwmSSJZR0YoRjFGMUYzRjEsJkY+RjFGMUYxRjFGPkYzLUY1NiRGN0YyRjFGMyowRjxGMUY7RjNGOUYxRj1GMUY/RjFGPkYzLUY1NiRGNyokRjJGM0YxRjEqKkYwRjFGOUYxRjJGMy1GNTYkRjJGN0YxRjMqJkYwRjEtLSZJIkRHRiU2I0YxNiNGNUY2RjFGMyoqRjxGMUY+RjFGMkYzSSJ0R0YoRjNGMTcjRi4=

n:=10: normal(series(subs(M(0,x)=subs(x=0,y=x,Mex(n)),M(0,1/x)=subs(x=0,y=1/x,Mex(n)), M(x,0)=subs(y=0,Mex(n)), (D[1](M))(0, 0)=coeff(subs(y=0,Mex(n)),x,1),M(0,0)=coeff(subs(y=0,Mex(n)),x,0),Y=Y0ser(n+2),f8),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

What we should remember from the above subsection f8;

LC4qKCIiJCIiIkkieEc2IiEiIi1JIk1HRic2JCIiIUYsRiVGJSosLCgqJClGJiIiI0YlRiVGJkYlRiVGJUYlLCZJIllHRidGJUYlRihGJSwmRjNGJUYlRiVGJUYzRigtRio2JEYsRiZGJUYoKjBGMUYlRjBGKEYuRiVGMkYlRjRGJUYzRigtRio2JEYsKiRGJkYoRiVGJSoqRiRGJUYuRiVGJkYoLUYqNiRGJkYsRiVGKComRiRGJS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjRiU2I0YqRitGJUYoKipGMUYlRjNGJUYmRihJInRHRidGKEYl

Rex:=proc(n) t^2*subs(y=0,Mex(n)): end: series(Rex(5),t,5);

KytJInRHNiIsJComIyIiIiIiJEYoSSJ4R0YkRihGKCIiIywmRihGKComRidGKClGKkYrRihGKEYpLCpGKUYoKiZGKUYoRipGKEYoKiYjRitGKUYoRi5GKEYoKiZGMkYoKUYqRilGKEYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiY=

Sex:=proc(n) t*x*subs(x=0,y=x,Mex(n)): end: series(Sex(5),t,5);

KytJInRHNiIqJkkieEdGJCIiIiwmRiZGJ0YnRidGJyIiIyomRiZGJywoKiZGKUYnKUYmRilGJ0YnKiZGKUYnRiZGJ0YnIiIkRidGJ0YvKiZGJkYnLCoqJiIiJ0YnKUYmRi9GJ0YnKiYiIzdGJ0YtRidGJyomIiNDRidGJkYnRiciIz9GJ0YnIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIiY=

Relation between R and S: S1=R0/t subs(x=0,Rex(6)); subs(x=0,diff(Sex(6),x));

KiYpSSJ0RzYiIiIjIiIiLC4qJiIlNVVGJylGJCIiJ0YnRicqJiIkeCdGJylGJCIiJkYnRicqJiIkMSJGJylGJCIiJUYnRicqJiIjP0YnKUYkIiIkRidGJyomRjhGJ0YjRidGJ0YkRidGJw==

KiZJInRHNiIiIiIsLiomIiU1VUYlKUYjIiInRiVGJSomIiR4J0YlKUYjIiImRiVGJSomIiQxIkYlKUYjIiIlRiVGJSomIiM/RiUpRiMiIiRGJUYlKiZGNkYlKUYjIiIjRiVGJUYjRiVGJQ==

We want to rewrite f8 in terms of R, S and Delta Introducing De=Delta f8form:=1/t*sqrt(De)*(x*M(0,x)-2/x*M(0,1/x))-3*(x+1/x+1)*M(x, 0)+2*Y/(x*t)-3*M10+3*M00/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdmOGZvcm1HRigsLCooSSJ0R0YoISIiKUkjRGVHRigjIiIiIiIjRjUsJiomSSJ4R0YoRjUtSSJNR0YoNiQiIiFGOUY1RjUqKEY2RjVGOUYxLUY7NiRGPSokRjlGMUY1RjFGNUY1KigiIiRGNSwoRjlGNUZBRjVGNUY1RjUtRjs2JEY5Rj1GNUYxKipGNkY1SSJZR0YoRjVGOUYxRjBGMUY1KiZGQ0Y1SSRNMTBHRihGNUYxKihGQ0Y1SSRNMDBHRihGNUY5RjFGNTcjRi4=

Let us check n:=10: normal(series(subs(De=Delta, M10=coeff(coeff(Mex(n),x,1),y,0),M00=coeff(coeff(Mex(n),x,0),y,0),M(0,1/x)=subs(x=0,y=1/x,Mex(n)), M(0,x)=subs(x=0,y=x,Mex(n)),M(x,0)=subs(y=0,Mex(n)), Y=Y0ser(n+2),f8form),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

eqRS:=collect(subs(M(0,x)=S(x)/t/x, M(x,0)=R(x)/t^2, M(0, 1/x)=S(1/x)/t*x,M00=R0/t^2, M10=R1/t^2,Y=(1-t*(x+1/x)-sqrt(De))/2/t/(1+x+1/x),t^2*f8form),[De,R,R1,S,R0],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVJTR0YoLCwqJiwoLUkiU0dGKDYjSSJ4R0YoIiIiKiYiIiNGNS1GMjYjKiRGNCEiIkY1RjsqJCwoKiQpRjRGN0Y1RjVGNEY1RjVGNUY7RjtGNSlJI0RlR0YoI0Y1RjdGNUY1KioiIiRGNUY9RjVGNEY7LUkiUkdGKEYzRjVGOyomRkRGNUkjUjFHRihGNUY7KihGREY1RjRGO0kjUjBHRihGNUY1KigsKComSSJ0R0YoRjVGP0Y1RjVGTkY1RjRGO0Y1RjRGO0Y9RjtGOzcjRi4=

The series Sh and Rh collect(subs(S(x)=Sh(x)+(-2*x-1/x)/(1+x+1/x)/3, S(1/x)=Sh(1/x)+(-2/x-x)/(1+x+1/x)/3, eqRS),[De,Sh,R,R1,R0],factor);

LCwqJiwmLUkjU2hHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwRiopSSNEZUdGJyNGKkYsRipGKioqIiIkRiosKCokKUYpRixGKkYqRilGKkYqRipGKkYpRjAtSSJSR0YnRihGKkYwKiZGNUYqSSNSMUdGJ0YqRjAqKEY1RipGKUYwSSNSMEdGJ0YqRioqKCwoKiZJInRHRidGKkY4RipGKkZCRipGKUYwRipGKUYwRjZGMEYw

ShS:=S(x)-(-2*x-1/x)/(1+x+1/x)/3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTaFNHRigsJi1JIlNHRig2I0kieEdGKCIiIiooI0YzIiIkRjMsJiomIiIjRjNGMkYzISIiKiRGMkY6RjpGMywoRjJGM0Y7RjNGM0YzRjpGOjcjRi4=

RhR:=-subs(De=0,eqRS);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSaFJHRigsKioqIiIkIiIiLCgqJClJInhHRigiIiNGMUYxRjVGMUYxRjFGMUY1ISIiLUkiUkdGKDYjRjVGMUYxKiZGMEYxSSNSMUdGKEYxRjEqKEYwRjFGNUY3SSNSMEdGKEYxRjcqKCwoKiZJInRHRihGMUY0RjFGMUZCRjFGNUY3RjFGNUY3RjJGN0YxNyNGLg==

The values of S(jj) and S(1/jj) jj:=exp(2*I*Pi/3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNqakdGKCwmIyIiIiIiIyEiIiomKiZGL0YwXiNGMEYwRjApIiIkRi9GMEYwNyNGLg==

op(1,simplify(factor(series(subs(De=Delta,eqRS),x=jj,3)),symbolic));

KiYsLCoqKiYiIiUiIiJeI0YnRidGJywmRihGJyokKSIiJCNGJyIiI0YnISIiRicsJkkidEc2IkYnRidGJ0YnLUkiU0dGMjYjLCQqJkYuRicsJiomRihGJ0YrRidGJ0YnRi9GL0YnRidGJyoqLCQqJkYuRidGKEYnRidGJ0YpRidGMEYnLUY0NiMsJkYtRi8qJiomRi1GJ0YoRidGJ0YrRidGJ0YnRi8qKEY8RicsJiomRixGJ0kjUjBHRjJGJ0YvKiZGLEYnSSNSMUdGMkYnRi9GJ0YrRidGJyomIiInRidGRUYnRicqJkZJRidGR0YnRi9GJyksJkYqRidGKEYnRi5GLw==

op(1,simplify(factor(series(subs(De=Delta,eqRS),x=1/jj,3)),symbolic));

LCwqJi1JIlNHNiI2IywkKiYiIiMiIiIsJiomXiNGK0YrKSIiJCNGK0YqRitGK0YrISIiRjJGK0YrLCZJInRHRiZGK0YrRitGK0YrKihGKkYrLUYlNiMsJkYxRjIqJiomRjFGK0YuRitGK0YvRitGK0YrRjNGK0YyKigqJiNGMEYqRitGLkYrRitGL0YrSSNSMEdGJkYrRisqJkY9RitGPkYrRjIqJkYwRitJI1IxR0YmRitGMg==

map(rationalize,solve({%%,%},{S(2/(I*sqrt(3)-1)),S(-1/2+I*sqrt(3)*(1/2))}));

PCQvLUkiU0c2IjYjLCQqJiIiIyIiIiwmKiZeI0YrRispIiIkI0YrRipGK0YrRishIiJGMkYrLCQqKiNGKyIiJUYrLCgqKComRipGK0YuRitGK0YvRitJI1IwR0YmRitGKyooKiZGMEYrRi5GK0YrRi9GK0kjUjFHRiZGK0YrKiZGMEYrRj1GK0YrRitGLEYrLCZJInRHRiZGK0YrRitGMkYrLy1GJTYjLCZGMUYyKiYqJkYxRitGLkYrRitGL0YrRissJCoqRjVGK0YsRissKiooRi5GK0YvRitGOkYrRitGO0YrKiZGMEYrRjpGK0YrRj5GK0YrRj9GMkYr

Not the same value... Sjsol:=op(2,op(2,%)); Sjjsol:=op(2,op(1,%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTanNvbEdGKCwkKiojIiIiIiIlRjEsJiomXiNGMUYxKSIiJCNGMSIiI0YxRjFGMSEiIkYxLCoqKEY1RjFGNkYxSSNSMEdGKEYxRjEqKComRjdGMUY1RjFGMUY2RjFJI1IxR0YoRjFGMSomRjdGMUY9RjFGMSomRjdGMUZARjFGMUYxLCZJInRHRihGMUYxRjFGOkYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdTampzb2xHRigsJCoqIyIiIiIiJUYxLCgqKComIiIjRjFeI0YxRjFGMSkiIiQjRjFGNkYxSSNSMEdGKEYxRjEqKComRjlGMUY3RjFGMUY4RjFJI1IxR0YoRjFGMSomRjlGMUY+RjFGMUYxLCYqJkY3RjFGOEYxRjFGMSEiIkYxLCZJInRHRihGMUYxRjFGQkYxNyNGLg==

eqRS;

LCwqJiwoLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjAqJCwoKiQpRilGLEYqRipGKUYqRipGKkYwRjBGKilJI0RlR0YnI0YqRixGKkYqKioiIiRGKkYyRipGKUYwLUkiUkdGJ0YoRipGMComRjlGKkkjUjFHRidGKkYwKihGOUYqRilGMEkjUjBHRidGKkYqKigsKComSSJ0R0YnRipGNEYqRipGQ0YqRilGMEYqRilGMEYyRjBGMA==

RHS2:=subs(De=0,eqRS)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVSSFMyR0YoKiQpLCoqKiIiJCIiIiwoKiQpSSJ4R0YoIiIjRjNGM0Y3RjNGM0YzRjNGNyEiIi1JIlJHRig2I0Y3RjNGOSomRjJGM0kjUjFHRihGM0Y5KihGMkYzRjdGOUkjUjBHRihGM0YzKigsKComSSJ0R0YoRjNGNkYzRjNGREYzRjdGOUYzRjdGOUY0RjlGOUY4RjM3I0Yu

negRHS2:=convert(map(factor,subs(R(0)=R0,D(R)(0)=R1,series(RHS2,x,2))),polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShuZWdSSFMyR0YoLCYqJilJInRHRigiIiMiIiIpSSJ4R0YoRjIhIiJGMyoqRjJGM0YxRjMsKiomIiInRjNJI1IxR0YoRjNGMyomIiIkRjNJI1IwR0YoRjNGM0YzRjZGMUY2RjNGNUY2RjM3I0Yu

If we want to go one step further, one needs R2 map(factor,subs(((D@@2)(R))(0)=2*R2,R(0)=R0,D(R)(0)=R1,series(RHS2,x,3)))

KytJInhHNiIqJClJInRHRiQiIiMiIiIhIiMsJCooRihGKUYnRiksKiomIiInRilJI1IxR0YkRilGKSomIiIkRilJI1IwR0YkRilGKUYpISIiRidGNEYpRilGNCw2KiYiIipGKSlGM0YoRilGKSooIiNPRilGM0YpRjBGKUYpKiZGOkYpKUYwRihGKUYpKihGL0YpRjBGKUYnRilGNCooRi9GKUkjUjJHRiRGKUYnRilGKSomRjJGKUYmRilGKSomRi9GKUYzRilGNComIiM3RilGMEYpRjQqJiIiJUYpRidGKUYpRilGKSIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YpRik=

LHS2:=subs(De=Delta,op(1,eqRS)^2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVMSFMyR0YoKiYpLCgtSSJTR0YoNiNJInhHRigiIiIqJiIiI0Y1LUYyNiMqJEY0ISIiRjVGOyokLCgqJClGNEY3RjVGNUY0RjVGNUY1RjtGO0Y3RjUsJiokKSwmRjVGNSomSSJ0R0YoRjUsJkY0RjVGOkY1RjVGO0Y3RjVGNSooIiIlRjUpRkVGN0Y1KSwoRjRGNUY6RjVGNUY1RjdGNUY7RjU3I0Yu

LHS2:=collect(LHS2,[S],distributed,factor);nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVMSFMyR0YoLC4qKiwqKiZJInRHRigiIiIpSSJ4R0YoIiIjRjNGMyooRjZGM0YyRjNGNUYzRjNGMkYzRjVGM0YzLCoqKCIiJEYzRjJGM0Y0RjNGM0Y3RjMqJkY6RjNGMkYzRjNGNSEiIkYzRjRGPCktSSJTR0YoNiNGNUY2RjNGPCouIiIlRjNGMEYzRjhGM0Y0RjxGPkYzLUY/NiMqJEY1RjxGM0YzKi5GNkYzRjBGM0Y4RjMsKCokRjRGM0YzRjVGM0YzRjNGPEY0RjxGPkYzRjMqLEZCRjNGMEYzRjhGM0Y0RjwpRkNGNkYzRjwqLkZCRjNGMEYzRjhGM0ZHRjxGNEY8RkNGM0Y8KipGMEYzRjhGMylGR0Y2RjxGNEY8Rjw3I0Yu

IiIn

Two terms are tricky LHS2special:=op(5,LHS2); LHS2mixed:=op(2,LHS2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxMSFMyc3BlY2lhbEdGKCwkKi4iIiUiIiIsKiomSSJ0R0YoRjEpSSJ4R0YoIiIjRjFGMSooRjdGMUY0RjFGNkYxRjFGNEYxRjZGMUYxLCoqKCIiJEYxRjRGMUY1RjFGMUY4RjEqJkY7RjFGNEYxRjFGNiEiIkYxLCgqJEY1RjFGMUY2RjFGMUYxRj1GNUY9LUkiU0dGKDYjKiRGNkY9RjFGPTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpMSFMybWl4ZWRHRigsJCouIiIlIiIiLCoqJkkidEdGKEYxKUkieEdGKCIiI0YxRjEqKEY3RjFGNEYxRjZGMUYxRjRGMUY2RjFGMSwqKigiIiRGMUY0RjFGNUYxRjFGOEYxKiZGO0YxRjRGMUYxRjYhIiJGMUY1Rj0tSSJTR0YoNiNGNkYxLUY/NiMqJEY2Rj1GMUYxNyNGLg==

The lemma (first part). The result should be x-negative k:=3: factor(jj^k/(1-jj)/(1-x*jj)+1/jj^k/(1-1/jj)/(1-x/jj)- x^(-k)/(1+x+x^2)): factor(numer(%)/expand(denom(%)));

KiYsJkkieEc2IiIiIkYmISIiRiYpRiQiIiRGJw==

Also works for k=-1 k:=-1: factor(jj^k/(1-jj)/(1-x*jj)+1/jj^k/(1-1/jj)/(1-x/jj)- x^(-k)/(1+x+x^2)): factor(numer(%)/expand(denom(%)));

IiIh

We focus on the "special" term. We want to extract from the numerator the terms involving powers of x larger than 1 map(factor,subs(S(0)=0,(D(S))(0)=R0/t,series(factor((1+x+x^2)*LHS2special),x=infinity,3))); posnumspecial:=op(1,%);

LCoqKiIjNyIiIkkidEc2IkYlSSNSMEdGJ0YlSSJ4R0YnRiUhIiIqKCIiJ0YlKUYmIiIjRiUtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiRJIkRHRjNGLjYjSSJTR0YnNiMiIiFGJUYqKigiIilGJSwmKiYiIiVGJUYmRiVGJUYlRiVGJUYoRiVGKi1JIk9HRjM2IyokRilGKkYl

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5wb3NudW1zcGVjaWFsR0YoLCQqKiIjNyIiIkkidEdGKEYxSSNSMEdGKEYxSSJ4R0YoRjEhIiI3I0Yu

negnumspecial:=factor((1+x+x^2)*LHS2special)-posnumspecial;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5uZWdudW1zcGVjaWFsR0YoLCYqLCIiJSIiIiwqKiZJInRHRihGMSlJInhHRigiIiNGMUYxKihGN0YxRjRGMUY2RjFGMUY0RjFGNkYxRjEsKiooIiIkRjFGNEYxRjVGMUYxRjhGMSomRjtGMUY0RjFGMUY2ISIiRjFGNUY9LUkiU0dGKDYjKiRGNkY9RjFGPSoqIiM3RjFGNEYxSSNSMEdGKEYxRjZGMUYxNyNGLg==

coeff(Delta,x,2);

LCQqJiIiJCIiIilJInRHNiIiIiNGJSEiIg==

With the notation of the lemma Fexpr:=map(factor,subs(x=1/x,negnumspecial));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZGZXhwckdGKCwmKiwiIiUiIiIsKiomSSJ0R0YoRjEpSSJ4R0YoIiIjRjFGMSooRjdGMUY0RjFGNkYxRjFGNEYxRjZGMUYxLCoqKCIiJEYxRjRGMUY1RjFGMUY4RjEqJkY7RjFGNEYxRjFGNiEiIkYxRjVGPS1JIlNHRig2I0Y2RjFGPSoqIiM3RjFGNEYxSSNSMEdGKEYxRjZGPUYxNyNGLg==

Fjexpr:=factor(subs(S(x)=Sjsol,x=jj,Fexpr));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdGamV4cHJHRigqJiwmKiZeIyIiIkYyKSIiJCNGMiIiI0YyRjJGNEYyRjIsMCoqRjFGMkYzRjJJI1IxR0YoRjJJInRHRihGMkYyKihGMUYyRjNGMkkjUjBHRihGMkYyKihGMUYyRjNGMkY5RjJGMiooIiIlRjJGPEYyRjpGMiEiIiooRjRGMkY5RjJGOkYyRkBGPEZAKiZGNEYyRjlGMkZARjI3I0Yu

Fjinvexpr:=factor(subs(S(x)=Sjjsol,x=1/jj,Fexpr));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpGamludmV4cHJHRigqJiwmKiZeIyIiIkYyKSIiJCNGMiIiI0YyRjJGNCEiIkYyLDAqKkYxRjJGM0YySSNSMUdGKEYySSJ0R0YoRjJGMiooRjFGMkYzRjJJI1IwR0YoRjJGMiooRjFGMkYzRjJGOkYyRjIqKCIiJUYyRj1GMkY7RjJGMiooRjRGMkY6RjJGO0YyRjJGPUYyKiZGNEYyRjpGMkYyRjI3I0Yu

factor(Fjexpr/(1-jj)/(1-x*jj)+Fjinvexpr/(1-1/jj)/(1-x/jj)): nonnegspecial:=factor(numer(%)/expand(denom(%)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5ub25uZWdzcGVjaWFsR0YoLCQqKCIiJSIiIiwuKioiIiNGMUkjUjBHRihGMUkidEdGKEYxSSJ4R0YoRjFGMSooRjRGMUY1RjFGNkYxISIiKiZGNUYxRjdGMUY5KigiIiRGMUkjUjFHRihGMUY2RjFGOSomRjRGMUY1RjFGOSomRjxGMUY9RjFGOUYxLCgqJClGN0Y0RjFGMUY3RjFGMUYxRjlGMTcjRi4=

negspecial:=collect(negnumspecial/(1+x+x^2)-nonnegspecial,S,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStuZWdzcGVjaWFsR0YoLCYqLiIiJSIiIiwqKiZJInRHRihGMSlJInhHRigiIiNGMUYxKihGN0YxRjRGMUY2RjFGMUY0RjFGNkYxRjEsKiooIiIkRjFGNEYxRjVGMUYxRjhGMSomRjtGMUY0RjFGMUY2ISIiRjEsKCokRjVGMUYxRjZGMUYxRjFGPUY1Rj0tSSJTR0YoNiMqJEY2Rj1GMUY9KipGMEYxLCZGNEYxRjFGMUYxLCgqJkkjUjBHRihGMUY2RjFGMSomRjdGMUZIRjFGMSomRjtGMUkjUjFHRihGMUYxRjFGPkY9RjE3I0Yu

Let us check. First, the above term should be a series with coefficients in 1/x Q[1/x] n:=10: factor(series(subs(S(1/x)=subs(x=1/x,Sex(n)), R0=subs(x=0,Rex(n)), R1=coeff(Rex(n),x,1), negspecial),t,n));

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

and then the difference with the special term should have non-negative powers of x factor(LHS2special-negspecial);

LCQqKiIiJSIiIiwmSSJ0RzYiRiVGJUYlRiUsKComSSNSMEdGKEYlSSJ4R0YoRiVGJSomIiIjRiVGK0YlRiUqJiIiJEYlSSNSMUdGKEYlRiVGJSwoKiQpRixGLkYlRiVGLEYlRiVGJSEiIkY1

Now we take each piece of LHS2, and extract the negative part res:=0: Since S(x) has no constant term in x, the first piece has negative part 0 op(1,LHS2);

LCQqKiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMSktSSJTR0YnNiNGKkYrRihGMQ==

For the second piece, we keep it as it is, calling its negative part as -4 NEG op(2,LHS2);factor(%/Delta);

LCQqLiIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiVGKkYyLUkiU0dGKTYjRitGJS1GNDYjKiRGK0YyRiVGJQ==

LCQqKCIiJSIiIi1JIlNHNiI2I0kieEdGKEYlLUYnNiMqJEYqISIiRiVGLg==

res:=res-4*NEG: Third piece: mostly positive op(3,LHS2);

LCQqLiIiIyIiIiwqKiZJInRHNiJGJSlJInhHRilGJEYlRiUqKEYkRiVGKEYlRitGJUYlRihGJUYrRiVGJSwqKigiIiRGJUYoRiVGKkYlRiVGLEYlKiZGL0YlRihGJUYlRishIiJGJSwoKiRGKkYlRiVGK0YlRiVGJUYxRipGMS1JIlNHRik2I0YrRiVGJQ==

subs((D(S))(0)=R0/t,S(0)=0, series(op(3,LHS2),x,2));

KydJInhHNiIsJCooIiInIiIiSSNSMEdGJEYoSSJ0R0YkRihGKCEiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIh

res:=res+convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJiomIiIlIiIiSSRORUdHRihGMSEiIioqIiInRjFJInRHRihGMUkjUjBHRihGMUkieEdGKEYzRjE3I0Yu

Fourth piece: mostly negative op(4,LHS2); subs((D(S))(0)=R0/t,S(0)=0,series(%,x=infinity,3));

LCQqLCIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiVGKkYyKS1JIlNHRik2IyokRitGMkYsRiVGMg==

LCYqJiIjNyIiIilJI1IwRzYiIiIjRiUhIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiMqJEkieEdGKEYqRiU=

res:=res+op(4,LHS2)-convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsKiomIiIlIiIiSSRORUdHRihGMSEiIioqIiInRjFJInRHRihGMUkjUjBHRihGMUkieEdGKEYzRjEqLEYwRjEsKiomRjZGMSlGOCIiI0YxRjEqKEY9RjFGNkYxRjhGMUYxRjZGMUY4RjFGMSwqKigiIiRGMUY2RjFGPEYxRjFGPkYxKiZGQUYxRjZGMUYxRjhGM0YxRjxGMyktSSJTR0YoNiMqJEY4RjNGPUYxRjMqJiIjN0YxKUY3Rj1GMUYxNyNGLg==

Now comes the tricky term op(5,LHS2);LHS2special;

LCQqLiIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiUsKCokRipGJUYlRitGJUYlRiVGMkYqRjItSSJTR0YpNiMqJEYrRjJGJUYy

LCQqLiIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiUsKCokRipGJUYlRitGJUYlRiVGMkYqRjItSSJTR0YpNiMqJEYrRjJGJUYy

res:=res+negspecial;

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

The sixth and last piece op(6,LHS2); factor(series(%,x,2));

LCQqKiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoKSwoKiRGKUYoRihGKkYoRihGKEYrRjFGKUYxRjE=

KylJInhHNiIsJComIiIkIiIiKUkidEdGJCIiI0YoISIiISIjLCQqKEYrRihGKkYoLCZGKkYoRihGKEYoRixGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIh

res:=res+convert(%,polynom);

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

In conclusion, extracting the negative part gives the following identity eqneg:=collect(res-negRHS2,[S,NEG,R1],factor);

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

collect(solve(eqneg,NEG),S,factor);

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

Let us ckeck: this should be the negative part of Delta S(x) S(1/x) n:=10: factor(series(subs(NEG=0,S(1/x)=subs(x=1/x,Sex(n)), R0=subs(x=0,Rex(n)), R1=coeff(Rex(n),x,1), eqneg/4),t,n));

KzFJInRHNiIqJEkieEdGJCEiIiIiJSwkKiYiIiQiIiJGJkYnRiwiIiYqJiwoKiYiIzhGLClGJiIiI0YsRiwqJkYoRixGJkYsRidGLEYnRiwpRiZGK0YnIiInKiYsKiomIiNpRixGNUYsRiwqJiIjSEYsRjJGLEYnKiYiIzZGLEYmRixGJ0YzRidGLClGJkYoRiciIigqJiwsKiYiJChHRixGP0YsRiwqJiIkbyNGLEY1RixGJyomIiQ2IkYsRjJGLEYnKiYiI0lGLEYmRixGJ0YtRidGLClGJkYtRiciIikqJiwuKiYiJSVcIkYsRktGLEYsKiYiJW4+RixGP0YsRicqJiIkTSpGLEY1RixGJyomIiQ6JEYsRjJGLEYnKiYiI3pGLEYmRixGJyIjN0YnRiwpRiZGNkYnIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiwiIzU=

Hence this should be x-nonnegative: all right! factor(series(Sex(n)*subs(x=1/x,Sex(n))*Delta-%,t,10));

KzFJInRHNiIsJkkieEdGJCIiIiIiI0YnIiIlLCYqJiIiJEYnRiZGJ0YnIiInRiciIiYsKiokKUYmRixGJyEiIiomRilGJylGJkYoRidGMiomIiM4RidGJkYnRiciI1RGJ0YtLCwqJkYoRicpRiZGKUYnRjIqJiIjNkYnRjFGJ0YyKiYiI0hGJ0Y0RidGMiomIiNpRidGJkYnRiciJDcjRiciIigsLiomRi5GJylGJkYuRidGMiomIiNJRidGOkYnRjIqJiIkNiJGJ0YxRidGMiomIiRvI0YnRjRGJ0YyKiYiJChHRidGJkYnRiciJW84RiciIiksMComIiM3RicpRiZGLUYnRjIqJiIjekYnRkVGJ0YyKiYiJDokRidGOkYnRjIqJiIkTSpGJ0YxRidGMiomIiVuPkYnRjRGJ0YyKiYiJSVcIkYnRiZGJ0YnIiVHJSlGJyIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiM1

Let us now reconstruct S(x)S(1/x) Delta. P0 is defined to be the constant term in x of this product eqS:=collect(-Delta*S(x)*S(1/x)+ subs(NEG=0,eqneg/4) + subs(NEG=0,x=1/x,eqneg/4)+P0 ,[S,P0],distributed,factor);

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

Of course we check: OK n:=10: factor(series(subs(S(x)=Sex(n),S(1/x)=subs(x=1/x,Sex(n)), R0=subs(x=0,Rex(n)), R1=coeff(Rex(n),x,1), eqS),t,n));

KzNJInRHNiJJI1AwR0YkIiIhISIjIiIlISInIiImISNUIiInISQ3IyIiKCElbzgiIikhJUclKSIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjNQ==

Several forms for the term with no S map(factor,convert(subs(S(x)=0,S(1/x)=0,eqS),parfrac,x));

LDgqJilJInRHNiIiIiMiIiIpSSJ4R0YmRidGKCEiIioqIiIkRihJI1IxR0YmRihGJUYoRipGKEYrKiYiIidGKClJI1IwR0YmRidGKEYoKihGJ0YoRjJGKEYlRihGKCooRi1GKEYuRihGJUYoRihJI1AwR0YmRigqJkYnRihGMkYoRigqJkYtRihGLkYoRigqKkYtRihGLkYoRiVGKEYqRitGKyosRi1GKEYuRigsJkYlRihGKEYoRihGKkYoLCgqJEYpRihGKEYqRihGKEYoRitGKyomRiRGKEYpRitGKw==

In terms of S-hat SSh:=collect(solve(Sh(x)-ShS,S(x)),Sh,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTU2hHRigsJi1JI1NoR0YoNiNJInhHRigiIiIqKCNGMyIiJEYzLCYqJiIiI0YzKUYyRjlGM0YzRjNGM0YzLCgqJEY6RjNGM0YyRjNGM0YzISIiRj03I0Yu

factor(Delta); collect(subs(S(x)=SSh,S(1/x)=subs(x=1/x,SSh),eqS),[Sh,NEG],distributed, factor);

LCQqKCwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMUYx

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

eqS;

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

P0ex:=proc(n) factor(series(Delta*Sex(n)*subs(x=1/x,Sex(n)),t,n+1)): coeff(convert(%,polynom),x,0); end:

ShS;

LCYtSSJTRzYiNiNJInhHRiUiIiIqKCNGKCIiJEYoLCYqJiIiI0YoRidGKCEiIiokRidGL0YvRigsKEYnRihGMEYoRihGKEYvRi8=

We multiply by Shat(x)+Shat(1/x) to decouple terms in S(x) and S(1/x) eqSdec:=collect(eqS*(ShS+subs(x=1/x,ShS)),[S],distributed,factor): nops(%);

IiIo

We want to extract the nonnegative part from this first term : x-positive res:=0: op(1,eqSdec);

LCQqKiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMSktSSJTR0YnNiNGKkYvRihGMQ==

res:=res+%;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCoqLCoqJkkidEdGKCIiIilJInhHRigiIiNGM0YzKihGNkYzRjJGM0Y1RjNGM0YyRjNGNUYzRjMsKiooIiIkRjNGMkYzRjRGM0YzRjdGMyomRjpGM0YyRjNGM0Y1ISIiRjNGNEY8KS1JIlNHRig2I0Y1RjpGM0Y8NyNGLg==

2nd term: non-negative in x op(2,eqSdec);

LCQqLiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMSwmKiZGK0YoRilGKEYoRihGKEYoLCgqJEYpRihGKEYqRihGKEYoRjEpLUkiU0dGJzYjRipGK0YoRjE=

res:=res+%: Third term : mostly positive op(3,eqSdec);

KiosaHAqKCIiJSIiIilJInRHNiIiIiNGJilJInhHRikiIilGJiEiIioqIiIkRiZJI1IxR0YpRiZGKEYmKUYsIiIoRiZGLiooIiM1RiZGJ0YmRjJGJkYuKigiIidGJilJI1IwR0YpRipGJilGLEY3RiZGJioqRipGJkY5RiZGKEYmRjpGJkYmKipGMEYmRjFGJkYoRiZGOkYmRi4qKCIjO0YmRidGJkY6RiZGLiooRipGJkYoRiZGMkYmRi4qJkkjUDBHRilGJkY6RiZGJiooIiM3RiZGOEYmKUYsIiImRiZGJioqRiVGJkY5RiZGKEYmRkRGJkYmKihGKkYmRjlGJkY6RiZGJioqIiIqRiZGMUYmRihGJkZERiZGLiooRjBGJkYxRiZGOkYmRiYqKCIjPUYmRidGJkZERiZGLiooRipGJkZBRiZGREYmRiYqKEZMRiZGOEYmKUYsRiVGJkYmKipGN0YmRjlGJkYoRiZGT0YmRiYqKEYlRiZGOUYmRkRGJkYmKipGN0YmRjFGJkYoRiZGT0YmRi4qKEYwRiZGMUYmRkRGJkYmKigiIzpGJkYnRiZGT0YmRi4qKEYlRiZGKEYmRkRGJkYuKiRGOkYmRiYqKEYwRiZGQUYmRk9GJkYmKihGQ0YmRjhGJilGLEYwRiZGJioqRiVGJkY5RiZGKEYmRlpGJkYmKihGN0YmRjlGJkZPRiZGJioqRklGJkYxRiZGKEYmRlpGJkYuKihGN0YmRjFGJkZPRiZGJiooRjVGJkYnRiZGWkYmRi4qKEYqRiZGQUYmRlpGJkYmKihGN0YmRjhGJilGLEYqRiZGJioqRipGJkY5RiZGKEYmRlxvRiZGJiooRiVGJkY5RiZGWkYmRiYqKkYwRiZGMUYmRihGJkZcb0YmRi4qKEYwRiZGMUYmRlpGJkYmKihGN0YmRidGJkZcb0YmRi4qKEYqRiZGKEYmRlpGJkYuKiRGT0YmRiYqJkZBRiZGXG9GJkYmKihGKkYmRjlGJkZcb0YmRiYqKkYwRiZGMUYmRihGJkYsRiZGLiooRjBGJkYxRiZGXG9GJkYmKihGKkYmRidGJkYsRiZGLiokRidGJkYuRiYpLCgqJEZcb0YmRiZGLEYmRiZGJkYqRi5GXG9GLi1JIlNHRik2I0YsRiY=

factor(subs((D(S))(0)=R0/t,S(0)=0,series(op(3,eqSdec),x,2)));

KydJInhHNiIsJComSSNSMEdGJCIiIkkidEdGJEYoISIiRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiIQ==

res:=res+op(3,eqSdec)-convert(%,polynom);

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

4th: x-negative, does not contribute op(4,eqSdec); subs(S(0)=0,series(%,x=infinity,3));

LCQqKiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMSktSSJTR0YnNiMqJEYqRjFGL0YoRjE=

LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiRJInhHRichIiI=

5th: a first tricky term op(5,eqSdec);

LCQqLiwqKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKCooRitGKEYmRihGKkYoRihGJkYoRipGKEYoLCoqKCIiJEYoRiZGKEYpRihGKEYsRigqJkYvRihGJkYoRihGKiEiIkYoRilGMSwmKiRGKUYoRihGK0YoRigsKEYzRihGKkYoRihGKEYxKS1JIlNHRic2IyokRipGMUYrRihGMQ==

num:=factor(%*(1+x+x^2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRudW1HRigsJCosLCoqJkkidEdGKCIiIilJInhHRigiIiNGM0YzKihGNkYzRjJGM0Y1RjNGM0YyRjNGNUYzRjMsKiooIiIkRjNGMkYzRjRGM0YzRjdGMyomRjpGM0YyRjNGM0Y1ISIiRjNGNEY8LCYqJEY0RjNGM0Y2RjNGMyktSSJTR0YoNiMqJEY1RjxGNkYzRjw3I0Yu

The terms in x^2 or more in num numpos:=convert(map(factor,subs(((D@@2)(S))(0)=2*S2,(D(S))(0)=R0/t,S(0)=0,series(num,x=infinity,3))),polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdudW1wb3NHRigsJCooIiIkIiIiKUkjUjBHRigiIiNGMSlJInhHRihGNEYxISIiNyNGLg==

res:=res+numpos/(1+x+x^2): Fexpr:=map(factor,subs(x=1/x,num-numpos));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZGZXhwckdGKCwmKiwsKiomSSJ0R0YoIiIiKUkieEdGKCIiI0YzRjMqKEY2RjNGMkYzRjVGM0YzRjJGM0Y1RjNGMywqKigiIiRGM0YyRjNGNEYzRjNGN0YzKiZGOkYzRjJGM0YzRjUhIiJGMylGNSIiJUY8LCYqJkY2RjNGNEYzRjNGM0YzRjMpLUkiU0dGKDYjRjVGNkYzRjwqKEY6RjMpSSNSMEdGKEY2RjNGNEY8RjM3I0Yu

The non-negative part in F(1/x)/(1+x+x^2) (first part of the lemma) factor(subs(S(x)=Sjsol,x=jj,Fexpr)/(1-jj)/(1-jj*x) +subs(S(x)=Sjjsol,x=1/jj,Fexpr)/(1-1/jj)/(1-1/jj*x)): factor(numer(%)/expand(denom(%)));

LCQqKCIiJCIiIiwqKihGJEYlKUkjUjFHNiIiIiNGJUkieEdGKkYlISIiKiQpSSNSMEdGKkYrRiVGJSooIiInRiVGMEYlRilGJUYlKiZGJEYlRihGJUYlRiUsKCokKUYsRitGJUYlRixGJUYlRiVGLUYl

res:=res+%: 6th: another tricky term, this time with a double pole at jj op(6,eqSdec);

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

We need the second part of the lemma The lemma (second part). The result should be x-negative k:=4: factor( 1/(1-jj)^2*(jj^k/(1-jj*x)^2+ k*jj^k/(1-jj*x))+1/(1-1/jj)^2*(1/jj^k/(1-1/jj*x)^2+ k/jj^k/(1-1/jj*x))+2/3*jj^k/(1-jj)/(1-x*jj)+2/3/jj^k/(1-1/jj)/(1-x/jj)- x^(-k)/(1+x+x^2)^2): factor(numer(%)/expand(denom(%)));

LCQqJiwqKiYiIiMiIiIpSSJ4RzYiIiIkRidGJyokKUYpRiZGJ0YnKiZGJkYnRilGJyEiIkYnRidGJylGKSIiJUYvRi8=

Also works for k=-1, -2 and -3 for k from -3 to -1 do factor( 1/(1-jj)^2*(jj^k/(1-jj*x)^2+ k*jj^k/(1-jj*x))+1/(1-1/jj)^2*(1/jj^k/(1-1/jj*x)^2+ k/jj^k/(1-1/jj*x))+2/3*jj^k/(1-jj)/(1-x*jj)+2/3/jj^k/(1-1/jj)/(1-x/jj)- x^(-k)/(1+x+x^2)^2): print(factor(numer(%)/expand(denom(%))));od:

IiIh

IiIh

IiIh

num:=factor((1+x+x^2)^2*op(6,eqSdec)): numpos:=convert(map(factor,subs(((D@@2)(S))(0)=2*S2,S(0)=0,D(S)(0)=R0/t,series(num,x=infinity,3))),polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdudW1wb3NHRigsJiooSSNSMEdGKCIiIkkidEdGKEYxKUkieEdGKCIiJkYxISIiKiYsKComSSNTMkdGKEYxKUYyIiIjRjFGMSooIiIkRjFGMEYxSSNSMUdGKEYxRjEqKEY8RjFGMEYxRjJGMUYxRjEpRjQiIiVGMUY2NyNGLg==

res:=res+numpos/(1+x+x^2)^2: numneg:=num-numpos: Fexpr:=subs(x=1/x,numneg): factor(1/(1-jj)^2*(subs(S(x)=Sjsol,x=jj,Fexpr)/(1-jj*x)^2+jj*subs(diff(S(x), x)=dSj/jj, S(x)=Sjsol,x=jj,diff(Fexpr,x))/(1-jj*x)) +1/(1-1/jj)^2*(subs(S(x)=Sjjsol,x=1/jj,Fexpr)/(1-1/jj*x)^2+1/jj*subs(diff(S(x), x)=dSjinv*jj, S(x)=Sjjsol,x=1/jj,diff(Fexpr,x))/(1-1/jj*x)) +2/3*(1/(1-jj)* subs(S(x)=Sjsol,x=jj,Fexpr)/(1-jj*x) +1/(1-1/jj)* subs(S(x)=Sjjsol,x=1/jj,Fexpr)/(1-1/jj*x))): collect(factor(numer(%)/expand(denom(%))),[dSj,dSjinv],factor);

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

This should be the nonnegative part of numneg/(1+x+x^2)^2 resnonneg:=map(Z->factor(numer(Z)/expand(denom(Z))),collect(subs(dSj=(B1+I*sqrt(3)*B2)/(1+t)^2,dSjinv=(B1-I*sqrt(3)*B2)/(1+t)^2,%),[B1,B2],factor));

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

B1ex:=proc(n) map(expand,map(rationalize,1/2*convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))+(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))): end: B1ex(4);

LCgqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomI0YnRiZGJylGKUYlRidGKyomIyIjOEYmRicpRikiIiVGJ0Yr

B2ex:=proc(n) collect(1/2/I/sqrt(3)*map(expand,map(rationalize,convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))-(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))),t): end: B2ex(4);

LCgqJiMiIigiIiMiIiIpSSJ0RzYiIiIlRichIiIqJiMiIiRGJkYnKUYpRi9GJ0YsKiYjRidGJkYnKUYpRiZGJ0Ys

n:=8: map(normal,series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))-B1ex(n)-I*sqrt(3)*B2ex(n),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

Let us check: this should be x negative n:=10:factor(series(subs(S(1/x)=subs(x=1/x,Sex(n)), B1=B1ex(n),B2=B2ex(n), R0=subs(x=0,Rex(n)),R1=coeff(Rex(n),x,1),S2=coeff(Sex(n),x,2),P0=P0ex(n),numneg/(1+x+x^2)^2-resnonneg),t,n));

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res:=res+resnonneg: 7th term op(7,eqSdec); factor(series(%,x,2));

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KylJInhHNiIsJCokKUkidEdGJCIiIyIiIiEiIiEiIywkKiZGKEYqLCYqJiIiJEYqSSNSMUdGJEYqRipGKEYrRipGK0YrLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRioiIiE=

res:=res+op(7,eqSdec)-convert(%,polynom): indets(res);

PCtJI0IxRzYiSSNCMkdGJEkjUDBHRiRJI1IwR0YkSSNSMUdGJEkjUzJHRiRJInRHRiRJInhHRiQtSSJTR0YkNiNGKw==

eqSa:=collect(res,[S,B1,B2,P0],factor);

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

Let us check: OK n:=16:factor(series(subs(S(x)=Sex(n), P0=P0ex(n),B1=B1ex(n),B2=B2ex(n), R0=subs(x=0,Rex(n)),R1=coeff(Rex(n),x,1),S2=coeff(Sex(n),x,2),eqSa),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzs=

There are many unknown series... can we relate them? indets(eqSa);

PCtJI0IxRzYiSSNCMkdGJEkjUDBHRiRJI1IwR0YkSSNSMUdGJEkjUzJHRiRJInRHRiRJInhHRiQtSSJTR0YkNiNGKw==

factor(subs(((D@@2)(S))(0)=2*S2,(D(S))(0)=R0/t,S(0)=0,series(eqSa, x,3)));

KydJInhHNiIsOComIiInIiIiKUkjUjBHRiQiIiNGKEYoKihGK0YoKUkidEdGJEYrRihJI1MyR0YkRighIiIqKCIiKkYoRipGKEkjUjFHRiRGKEYoKiYjRigiIiRGKEkjQjFHRiRGKEYwSSNCMkdGJEYwSSNQMEdGJEYoKigjIiM5RjZGKEYqRihGLkYoRjAqJkYyRigpRjNGK0YoRigqKEY2RihGM0YoRi5GKEYwKiRGLUYoRjAqJiNGK0Y2RihGKkYoRjAiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

P0S2sol:=solve(op(1,%),P0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShQMFMyc29sR0YoLDYqJiIiJyIiIilJI1IwR0YoIiIjRjEhIiIqKEY0RjEpSSJ0R0YoRjRGMUkjUzJHRihGMUYxKigiIipGMUYzRjFJI1IxR0YoRjFGNSomI0YxIiIkRjFJI0IxR0YoRjFGMUkjQjJHRihGMSooIyIjOUY/RjFGM0YxRjhGMUYxKiZGO0YxKUY8RjRGMUY1KihGP0YxRjxGMUY4RjFGMSokRjdGMUYxKiYjRjRGP0YxRjNGMUYxNyNGLg==

eqSb:=collect(subs(P0=P0S2sol,eqSa),[S,B1,B2,R1,S2],factor);

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

Expanding further around x=0 ? not informative factor(subs(((D@@2)(S))(0)=2*S2,(D(S))(0)=R0/t,S(0)=0,series(eqSb, x,4)));

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

Expansion around jj factor(subs((D(S))(-1/2+I*sqrt(3)*(1/2))=(B1+I*sqrt(3)*B2)/(1+t)^2/jj,S(-1/2+I*sqrt(3)*(1/2))=Sjsol,series(eqSb,x=jj,2)));

KycsKEkieEc2IiIiIiNGJiIiI0YmKiYsJComRidGJl4jRiZGJkYmRiYpIiIkRidGJiEiIiwkKigjRiYiIipGJiwmKiZGLEYmRi1GJkYmRi5GJkYmLCwqKEYuRiZJI1MyR0YlRiYpSSJ0R0YlRihGJkYmKigiIihGJkkjUjBHRiVGJkY6RiZGJiomRi5GJkY5RiZGJiomRihGJkkjQjFHRiVGJkYmRj1GJkYmRi9GLy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIh

S2sol:=solve(op(1,%),S2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTMnNvbEdGKCwkKigjIiIiIiIkRjEsKiooIiIoRjFJI1IwR0YoRjFJInRHRihGMUYxKiZGMkYxKUY3IiIjRjFGMSomRjpGMUkjQjFHRihGMUYxRjZGMUYxRjkhIiJGPTcjRi4=

P0sol:=factor(subs(S2=S2sol,P0S2sol));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZQMHNvbEdGKCwwKiYiIiciIiIpSSNSMEdGKCIiI0YxISIiKigiIipGMUYzRjFJI1IxR0YoRjFGNSomRjdGMSlGOEY0RjFGNSooIiIkRjFGOEYxSSJ0R0YoRjFGMSokKUY9RjRGMUY1SSNCMUdGKEY1SSNCMkdGKEYxNyNGLg==

eqSc:=collect(subs(S2=S2sol,eqSb),[S,B1,B2,R1],factor);

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

Expansion around 1/jj: brings nothing new factor(subs((D(S))(2/(I*sqrt(3)-1))=(B1-I*sqrt(3)*B2)/(1+t)^2*jj,S(2/(I*sqrt(3)-1))=Sjjsol,series(eqSc,x=1/jj,0)));

KyUsJkkieEc2IiIiIiokLCYjRiYiIiMhIiIqJiomRilGJl4jRiZGJkYmKSIiJEYpRiZGJkYrRistSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJiIiIQ==

We have reached an equation with one catalytic variable for S(x) eqScat:=eqSc: collect(eqScat-Delta*ShS^3,[S,B1,B2,R1],factor): n:=20:factor(series(subs(S(x)=Sex(n), B1=B1ex(n),B2=B2ex(n), R0=subs(x=0,Rex(n)),R1=coeff(Rex(n),x,1),eqScat),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

indets(eqScat)

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSJ0R0YkSSJ4R0YkLUkiU0dGJDYjRik=

We have reached an equation with one catalytic variable for S(x) eqScat:=-(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)*S(x)^3/x^2-(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)*(2*x^2+1)*S(x)^2/(x^2*(x^2+x+1))+(-B1+B2-9*R1^2-(3*(t*x^4+3*R0*x^3-t*x^3+3*R0*x^2+t*x^2-x^3+3*R0*x-t*x+t-x))*R1/(x*(x^2+x+1))+(-4*t^2*x^8-10*t^2*x^7+2*R0*t*x^6-17*t^2*x^6-2*t*x^7+4*R0*t*x^5+2*R0*x^6-20*t^2*x^5+6*R0*t*x^4+4*R0*x^5-18*t^2*x^4-4*t*x^5+x^6+4*R0*t*x^3+6*R0*x^4-12*t^2*x^3+2*R0*t*x^2+4*R0*x^3-7*t^2*x^2-2*t*x^3+x^4+2*R0*x^2-2*t^2*x-t^2)/((x^2+x+1)^2*x^2))*S(x)-(x^2+2)*B1/(3*(x^2+x+1))+x^2*B2/(x^2+x+1)-9*x^2*R1^2/(x^2+x+1)-(3*(t*x^5+4*R0*x^4-t*x^4+4*R0*x^3+t*x^3-x^4+3*R0*x^2-t*x^2-R0*x+t*x-x^2-R0))*R1/(x^2+x+1)^2-(3*t^2*x^7+3*R0*t*x^6+3*t^2*x^6+9*R0^2*x^5-7*R0*t*x^5+6*t^2*x^5+9*R0^2*x^4-4*R0*t*x^4-7*R0*x^5+3*t^2*x^4-6*R0*t*x^3-4*R0*x^4+6*t^2*x^3-9*R0^2*x^2-2*R0*t*x^2-6*R0*x^3+3*t^2*x^2-9*R0^2*x+R0*t*x-2*R0*x^2+3*t^2*x-3*R0*t+R0*x)/(3*(x^2+x+1)^2*x)

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

indets(eqScat)

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSJ0R0YkSSJ4R0YkLUkiU0dGJDYjRik=

We work with S-hat, because then there is no quadratic term SSh:=isolate(Sh-ShS,S(x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTU2hHRigvLUkiU0dGKDYjSSJ4R0YoLCZJI1NoR0YoIiIiKigjRjUiIiRGNSwmKiYiIiNGNUYyRjUhIiIqJEYyRjxGPEY1LChGMkY1Rj1GNUY1RjVGPEY1NyNGLg==

eqShcat:=collect(subs(SSh,eqScat),[Sh,R0],factor): coeff(%,Sh,2)

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This equation is almost symmetric in x and 1/x factor(coeff(eqShcat,Sh,3)/subs(x=1/x,coeff(eqShcat,Sh,3))); factor(coeff(eqShcat,Sh,1)/subs(x=1/x,coeff(eqShcat,Sh,1))); factor(coeff(eqShcat,Sh,0)/subs(x=1/x,coeff(eqShcat,Sh,0)))

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This suggests to write Sh=(x-1/x) Sb, for a new seris Sb (stands for S-bar) eqSbcat:=numer(factor(subs(Sh=(x-1/x)*Sb,eqShcat/(x-1/x)))): nops(%);

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factor(coeff(eqSbcat,Sb,3)/subs(x=1/x,coeff(eqSbcat,Sb,3))); factor(coeff(eqSbcat,Sb,1)/subs(x=1/x,coeff(eqSbcat,Sb,1))); factor(coeff(eqSbcat,Sb,0)/subs(x=1/x,coeff(eqSbcat,Sb,0)))

KiQpSSJ4RzYiIiM5IiIi

KiQpSSJ4RzYiIiM5IiIi

KiQpSSJ4RzYiIiM5IiIi

Now we introduce y=x+1/x+1 We want to write eqSbcat/x^7 as a polynomial in 1+x+1/x=y res:=eqSbcat/x^7:algy:=0: for i from 0 to 7 do algy:=algy+y^(7-i)*coeff(res,x,7-i): res:= expand(res-(1+x+1/x)^(7-i)*coeff(res,x,7-i)) od: factor(subs(y=1+x+1/x,algy)/eqSbcat);

KiQpSSJ4RzYiIiIoISIi

eqSbcaty:=collect(algy,Sb,factor): factor(%); nops(%);

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degree(eqSbcaty,Sb);degree(eqSbcaty,y);

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collect(eqSbcaty,Sb,factor)

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

ldegree(coeff(eqSbcaty,Sb,3),y); ldegree(coeff(eqSbcaty,Sb,1),y); ldegree(coeff(eqSbcaty,Sb,0),y);

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This tells us to write Sb=St/y, hence Sh=(x-1/x)St/(x+1/x+1). This St is the series tilde_S in the paper. eqStcaty:=collect(subs(Sb=St/y,eqSbcaty),St,factor); degree(%,y)

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

IiIl

Another invariance property, now in y co3:=coeff(eqStcaty,St,3);collect(%,y,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjbzNHRigsJCosIiNGIiIiLCZJInlHRihGMUYxRjFGMSwmRjNGMSIiJCEiIkYxLCgqJkkidEdGKEYxRjNGMUYxRjlGMUYxRjFGMSwoKihGNUYxRjlGMUYzRjFGMUY5RjZGMUY2RjFGNjcjRi4=

LCwqKCIjIikiIiIpSSJ5RzYiIiIlRiUpSSJ0R0YoIiIjRiUhIiIqKiIjYUYlRitGJSwmKiZGLEYlRitGJUYlRiVGLUYlKUYnIiIkRiVGJSomKUYnRixGJSwoKiYiJHkkRiVGKkYlRiUqJiIkaSJGJUYrRiVGJSIjRkYlRiVGJSoqRi9GJSwmRitGJUYlRiVGJUYwRiVGJ0YlRiUqJkYkRiUpRj1GLEYlRi0=

factor(subs(y=(t+1)/t/y,co3)/co3)

KigpLCZJInRHNiIiIiJGJ0YnIiIjRicpSSJ5R0YmIiIlISIiKUYlRihGLA==

co1:=coeff(eqStcaty,St,1):collect(%,y,factor)

LCgqKiIiKiIiIkkidEc2IkYlLCgqJkYkRiVJI1IxR0YnRiVGJSomIiIjRiVGJkYlISIiRixGLUYlKUkieUdGJyIiJEYlRi0qJiw4KigiJFYjRiVJI1IwR0YnRiVGKkYlRi0qKCIjYUYlRjVGJUYmRiVGJSomRjRGJSlGKkYsRiVGLSooRjRGJUYqRiVGJkYlRiUqJiIjWEYlKUYmRixGJUYtKiYiI0ZGJUkjQjFHRidGJUYtKiZGP0YlSSNCMkdGJ0YlRiUqJkY3RiVGNUYlRiUqJiIjIilGJUYqRiVGJSomRjdGJUYmRiVGLUYkRi1GJSlGL0YsRiVGJSoqRiRGJSwmRiZGJUYlRiVGJUYoRiVGL0YlRi0=

factor(subs(y=(t+1)/t/y,co1)/co1)

KigpLCZJInRHNiIiIiJGJ0YnIiIjRicpSSJ5R0YmIiIlISIiKUYlRihGLA==

co0:=coeff(eqStcaty,St,0):collect(%,y,factor)

LCwqKCIiJCIiIilJInlHNiIiIiVGJSlJInRHRigiIiNGJSEiIiooRitGJSwqKiYiI0ZGJUkjUjBHRihGJUYlKiZGMUYlSSNSMUdGKEYlRiUqJiIjNUYlRitGJUYtRilGLUYlKUYnRiRGJUYtKiYpRidGLEYlLDoqJiIjIilGJSlGMkYsRiVGLSooIiRpIkYlRjJGJUY0RiVGLSooRjxGJUYyRiVGK0YlRiUqJkY8RiUpRjRGLEYlRi0qKEY8RiVGNEYlRitGJUYlKiRGKkYlRi0qJiIiKkYlSSNCMUdGKEYlRiUqJkZGRiVJI0IyR0YoRiVGJUYwRiVGM0YlKiYiIzdGJUYrRiVGLUYsRi1GJUYlKigsJkYrRiVGJUYlRiVGL0YlRidGJUYtKiZGJEYlKUZNRixGJUYt

factor(subs(y=(t+1)/t/y,co0)/co0)

KigpLCZJInRHNiIiIiJGJ0YnIiIjRicpSSJ5R0YmIiIlISIiKUYlRihGLA==

We introduce z=t*y+(t+1)/y res:=eqStcaty/y^(2):algz:=0: for i from 0 to 2 do algz:=algz+(z/t)^(2-i)*coeff(res,y,2-i): res:= expand(res-(y+(t+1)/t/y)^(2-i)*coeff(res,y,2-i)) od: factor(subs(z=t*y+(t+1)/y,algz)/eqStcaty);

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eqStcatz:=collect(algz,[St,R0],factor); nops(factor(%)); indets(%%)

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IiNQ

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Let us check factor(subs(St=1/(x-1/x)*((x+1/x+1)*S(x)+2*x/3+1/x/3), z=t*(x+1+1/x)+(t+1)/(x+1+1/x), eqStcatz)/eqScat)

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eqStcatz:=(27*(2*t+z+1))*(10*t-3*z+1)*St^3+((54-243*R1+54*t)*R0-243*R1^2+243*R1*t-81*z*R1-45*t^2+18*z*t-27*B1+27*B2+81*R1-54*t+18*z-9)*St-81*R0^2+(81*t+27-27*z-162*R1)*R0-81*R1^2+81*R1*t-27*z*R1+5*t^2+10*z*t-3*z^2+9*B1+9*B2+27*R1-6*t+4*z-2;

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

collect(%,[St,R0],factor)

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latex(%)

27\134, \134left( 2\134,t+z+1 \134right) \134left( 10\134,t-3\134,z+1 \134right) {{\134it St}}^{3}+ \134left( \134left( 54-243\134,{\134it R1}+54\134,t \134right) {\134it R0}-243\134,{{\134it R1}}^{2}+243\134,{\134it R1}\134,t-81\134,z{\134it R1}-45\134,{t}^{2}+18\134,zt-27\134,{\134it B1 }+27\134,{\134it B2}+81\134,{\134it R1}-54\134,t+18\134,z-9 \134right) {\134it St}-81\134,{{\134it R0}}^{2}+ \134left( 81\134,t+27-27\134,z-162\134,{\134it R1} \134right) {\134it R0}-81\134,{{ \134it R1}}^{2}+81\134,{\134it R1}\134,t-27\134,z{\134it R1}+5\134,{t}^{2}+10\134,zt-3\134,{z}^{2 }+9\134,{\134it B1}+9\134,{\134it B2}+27\134,{\134it R1}-6\134,t+4\134,z-2

zx:=t*(x+1+1/x)+(t+1)/(x+1+1/x); StS:=1/(x-1/x)*((x+1/x+1)*S(x)+2*x/3+1/x/3)

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTdFNHRigqJiwmSSJ4R0YoIiIiKiRGMCEiIkYzRjMsKComLChGMEYxRjJGMUYxRjFGMS1JIlNHRig2I0YwRjFGMSomIyIiIyIiJEYxRjBGMUYxKiYjRjFGPUYxRjBGM0YxRjE3I0Yu

B1ex:=proc(n) map(expand,map(rationalize,1/2*convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))+(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))): end: B1ex(4);

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B2ex:=proc(n) collect(1/2/I/sqrt(3)*map(expand,map(rationalize,convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))-(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))),t): end: B2ex(4);

LCgqJiMiIigiIiMiIiIpSSJ0RzYiIiIlRichIiIqJiMiIiRGJkYnKUYpRi9GJ0YsKiYjRidGJkYnKUYpRiZGJ0Ys

R0ex:=proc(n) coeff(Rex(n),x,0): end: R1ex:=proc(n) coeff(Rex(n),x,1): end: n:=22: normal(series(subs(S(x)=Sex(n), R0=R0ex(n),R1=R1ex(n),B1=B1ex(n),B2=B2ex(n),eqScat),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0I=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=