This is Part 1 of the Maple files that accompany the paper "Walks avoiding a quadrant and the reflection principle" by Mireille Bousquet-M\303\251lou and Michael Wallner. #############################################################Abstract:We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all 8 nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. We expect a similar algebraicity phenomenon to hold for the 7 Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for 3 of them. For the remaining 4, we predict the D-finite part of the solution, and in 3 of the 4 cases, give evidence for the algebraicity of the remaining part.############################################################# 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;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsLComLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxKiRGNSEiIkYxSSJ5R0YoRjEqJEY4RjdGMSomLCZGNUYxRjZGMUYxLCZGOEYxRjlGMUYxRjFGMUY3RjEtSSJDR0YoNiRGNUY4RjFGN0YxRjEqKkYzRjFGOEY3LChGNUYxRjZGMUYxRjFGMS1JI0NtR0YoNiNGNkYxRjcqKkYzRjFGNUY3LChGOEYxRjlGMUYxRjFGMS1GQzYjRjlGMUY3KipGM0YxRjVGN0Y4RjdJJEMwMEdGKEYxRjc3I0Yu

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)),eqC+C(x,y)),t,n+1)): fi: end: 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));

K0NJInRHNiIiIiIiIiEiIihGJSIjXSIiIyIkcCQiIiQiJXNGIiIlIiY/NSMiIiYiJ08yOyIiJyIoJmVON0YnIihjPGEqIiIpIilTdCRSKCIiKiIqKTMrWWQiIzUiK1NJMHdXIiM2IiwpMyJ5UVwkIiM3Ii1hNyFSPnQjIiM4Ii4jKipIPVFSQCIjOSIvRjBzXGJ4OyIjOg==

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

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);

NjIiIiIiIiEiIigiIz0iJGEiIiR3KCIldmAiJm1MJCInWXNBIihTTV8iIilTKVwwIiIpKVFjTCgiKlUtdjwmIisvRiEqek8iLHgiXCoqUUUiLSNbNCdlLz4=

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;

LCwqJiwmIiIiRiUqJkkidEc2IkYlLCxJInhHRihGJSokRiohIiJGJUkieUdGKEYlKiRGLUYsRiUqJiwmRipGJUYrRiVGJSwmRi1GJUYuRiVGJUYlRiVGLEYlLUkiQ0dGKDYkRipGLUYlRixGJUYlKipGJ0YlRi1GLCwoRipGJUYrRiVGJUYlRiUtSSNDbUdGKDYjRitGJUYsKipGJ0YlRipGLCwoRi1GJUYuRiVGJUYlRiUtRjg2I0YuRiVGLCoqRidGJUYqRixGLUYsSSRDMDBHRihGJUYs

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,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],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;

LCwqKiw0KihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKihGJkYoRilGKEYtRihGKCooRiZGKEYqRihGLEYoRigqJkYmRihGKUYoRigqJkYmRihGLEYoRigqJkYmRihGKkYoRigqJkYmRihGLUYoRigqJkYqRihGLUYoISIiRiZGKEYoRipGNUYtRjUtSSJBR0YnNiRGKkYtRihGKCosRiZGKCwoKiRGKUYoRihGKkYoRihGKEYoRipGNUYtRjUtSSNBbUdGJzYjKiRGKkY1RihGNSosRiZGKCwoKiRGLEYoRihGLUYoRihGKEYoRipGNUYtRjUtRj02IyokRi1GNUYoRjUqKkYmRihGKkY1Ri1GNUkkQTAwR0YnRihGNSoqI0YoIiIkRigsKCooRitGKEYpRihGLEYoRihGO0YoRkJGKEYoRilGNUYsRjVGKA==

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: n:=16: normal(series(subs(A00=A00ex(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));

LDYqJiIpaVppSSIiIilJInRHNiIiIzVGJUYlKiYiKEdMUyVGJSlGJyIiKkYlRiUqJiInSU1sRiUpRiciIilGJUYlKiYiJm9fKkYlKUYnIiIoRiVGJSomIiY/XyJGJSlGJyIiJ0YlRiUqJiIlb0BGJSlGJyIiJkYlRiUqJiIkQyVGJSlGJyIiJUYlRiUqJiIjW0YlKUYnIiIkRiVGJSomIiM9RiUpRiciIiNGJUYlRklGJQ==

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);

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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) (by symmetry the negative part of A(x,y) in y is M(1/y,x)) 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);

LCgqJiw6KiYiIiMiIiIpSSJ5RzYiRiZGJ0YnKiZGJkYnRilGJ0YnRiZGJyooIyIiJiIiJEYnSSJ4R0YqRidGKEYnRicqKCMiIilGL0YnRjBGJ0YpRidGJyomRi9GJ0YwRidGJyooI0YmRi9GJylGMEYmRidGKEYnRicqKEY2RidGN0YnRilGJ0YnKiZGNkYnRjdGJ0YnKigjRidGL0YnKUYwRi9GJ0YoRidGJyooRjZGJ0YpRidGPEYnRicqJkY2RidGPEYnRidGJylJInRHRipGJkYnRicqJiwsRilGJ0YnRicqKEY7RidGMEYnRilGJ0YnKihGO0YnRjdGJ0YpRidGJyomRjtGJ0Y3RidGJ0YnRkBGJ0YnKiZGO0YnRjBGJ0Yn

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))); 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 - well, in fact we will not use it. 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 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);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcU0xR0YoLDAqKCwmIiIiRjEqJkkidEdGKEYxLCxJInhHRihGMSokRjUhIiJGMUkieUdGKEYxKiRGOEY3RjEqJiwmRjVGMUY2RjFGMSwmRjhGMUY5RjFGMUYxRjFGN0YxRjVGNy1JIk1HRig2JEY2RjhGMUY3KiYjRjEiIiRGMSlGNSIiI0Y3RjEqLEYzRjEsKEY1RjFGNkYxRjFGMUYxRjhGN0Y1RjctRj42JEY2IiIhRjFGNyooRjNGMSwoRjhGMUY5RjFGMUYxRjEtRj42JEZJRjhGMUY3KihGM0YxRjhGNy1GPjYkRklGSUYxRjEqKkYzRjFGNUY3RktGMS1JIlBHRihGTUYxRjEqKkYzRjFGNUY3RjhGNy1GU0ZQRjFGNzcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcU0yR0YoLDAqKCwmIiIiRjEqJkkidEdGKEYxLCxJInhHRihGMSokRjUhIiJGMUkieUdGKEYxKiRGOEY3RjEqJiwmRjVGMUY2RjFGMSwmRjhGMUY5RjFGMUYxRjFGN0YxRjVGMS1JIk1HRig2JEY1RjhGMUY3KiYjRjEiIiRGMSlGNSIiI0YxRjEqLEYzRjEsKEY1RjFGNkYxRjFGMUYxRjhGN0Y1RjEtRj42JEY1IiIhRjFGNyooRjNGMSwoRjhGMUY5RjFGMUYxRjEtRj42JEZJRjhGMUY3KihGM0YxRjhGNy1GPjYkRklGSUYxRjEqKkYzRjFGNUYxRktGMS1JIlBHRihGTUYxRjEqKkYzRjFGNUYxRjhGNy1GU0ZQRjFGNzcjRi4=

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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

The equation in the paper eqM3pap:= -K*M(x,y)+x/3-t/y*(x+1/x+1)*M(x,0)-t/x*(y+1/y+1)*M(0,y)+t/x/y*M(0,0)+t*(y+1+1/y)*(limit(diff(M(x,y),x),x=0)+1/y*(M(y,0)-M(0,0)))-2*t/y*(D[1](M))(0, 0);

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

factor(eqM3-eqM3pap);

IiIh

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

KydJInhHNiIqJiw8KigtSSJNR0YkNiQiIiFJInlHRiQiIiJJInRHRiRGLSlGLCIiJEYtRi0qKiIiI0YtLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLTYjRilGKkYtRi5GLUYvRi1GLSoqRjJGLUYzRi1GLkYtKUYsRjJGLUYtKigtRik2JEYsRitGLUYuRi1GPUYtRi0qKC1GKTYkRitGK0YtRi5GLUY9Ri0hIiIqKEYoRi1GLkYtRixGLUYtKiZGKEYtRj1GLUZEKipGMkYtRi5GLUYzRi1GLEYtRi0qKEY/Ri1GLkYtRixGLUYtKipGMkYtRi5GLUZCRi1GLEYtRkQqKkYwRi1GLkYtLUY0RkNGLUYsRi1GRComRi5GLUY/Ri1GLSomRi5GLUZCRi1GREYtRj1GREYrLUkiT0dGN0Y6Ri0=

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

LCwqKkkidEc2IiIiIiksJkkieUdGJUYmRiZGJiIiI0YmKUYpRiohIiItSSJNR0YlNiQiIiFGMEYmRiwqKCwoKiZGJEYmRitGJkYmRiRGJkYpRixGJkYpRiwtRi42JEYwRilGJkYmKipGJEYmLCgqJEYrRiZGJkYpRiZGJkYmRiZGK0YsLUYuNiRGKUYwRiZGJioqIiIkRiZGJEYmRilGLC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiY2I0YuRi9GJkYsKixGKkYmRiRGJkY3RiZGKUYsLUY+RjVGJkYm

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

Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIiNiNJIk1HRio2JCIiIUkieUdGKiwkKiwjRiwiIiNGLCwqKipJInRHRipGLCksJkYxRixGLEYsRjVGLClGMUY1ISIiLUYuNiRGMEYwRixGLCooLCgqJkY4RixGO0YsRixGOEYsRjFGPEYsRjFGPC1GLkYvRixGPCoqRjhGLCwoKiRGO0YsRixGMUYsRixGLEYsRjtGPC1GLjYkRjFGMEYsRjwqKiIiJEYsRjhGLEYxRjwtRiRGPkYsRixGLEY4RjxGREY8RjFGLEYs

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcU1HRigsMCouIyIiIiIiI0YxSSJ0R0YoRjEsKComSSJ4R0YoRjEpSSJ5R0YoRjJGMUYxRjZGMSomRjJGMUY4RjEhIiJGMUY2RjpGN0Y6LUkiTUdGKDYkIiIhRj5GMUY6KixGMEYxLC4qKEYzRjFGNkYxRjdGMUYxKihGMkYxRjNGMUY3RjFGMSomRjNGMUY2RjFGMSooRjJGMUYzRjFGOEYxRjEqJkY2RjFGOEYxRjoqJkYyRjFGM0YxRjFGMUY2RjpGOEY6LUY8NiRGPkY4RjFGOiosRjNGMSwoKiQpRjZGMkYxRjFGNkYxRjFGMUYxRjZGOkY4RjotRjw2JEY2Rj5GMUY6KiosNCooRjNGMUZMRjFGN0YxRjEqKEYzRjFGTEYxRjhGMUYxRkFGMSomRjNGMUZMRjFGMSomRjNGMUY3RjFGMUZDRjEqJkYzRjFGOEYxRjFGRUY6RjNGMUYxRjZGOkY4RjotRjw2JEY2RjhGMUYxKixGMEYxRjNGMSwoKiRGN0YxRjFGOEYxRjFGMUYxRjdGOi1GPDYkRjhGPkYxRjEqKkYwRjFGM0YxRjhGOi0tJkkiREdGJTYjRjE2I0Y8Rj1GMUY6KiYjRjEiIiRGMUY2RjFGMTcjRi4=

We have recovered the functional equation stated in the main theorem eqMpap:=-K*(2*M(x,y)-M(0,y))+2*x/3-2*t/y*(1+x+1/x)*M(x,0)+t/y*(y+1+1/y)*M(y,0)+t*(x-1/x)*(y+1+1/y)*M(0,y)-t*(1+1/y^2-2/x/y)*M(0,0)-t/y*(D[1](M))(0, 0);

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

factor(eqM-1/2*eqMpap);

IiIh

What we need to remember eqM:=-t*(x*y^2+x-2*y)*M(0, 0)/(2*x*y^2)-(t*x*y^2+2*t*y^2+t*x+2*t*y-x*y+2*t)*M(0, y)/(2*x*y)-t*(x^2+x+1)*M(x, 0)/(x*y)+(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)*M(x, y)/(x*y)+t*(y^2+y+1)*M(y, 0)/(2*y^2)-t*(D[1](M))(0, 0)/(2*y)+(1/3)*x;

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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+SSNmMUdGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwoKiYpSSJZR0YoRjJGMUkieEdGKEYxRjEqJkYyRjFGN0YxISIiRjhGMUYxRjhGOkY2RjotSSJNR0YoNiQiIiFGPkYxRjoqMkYwRjFGM0YxLCgqJEY2RjFGMUY3RjFGMUYxRjEsJkY4RjFGMUY6RjEsJkY4RjFGMUYxRjFGOEY6RjdGOi1GPDYkRj5GN0YxRjEqLEYzRjEsKCokKUY4RjJGMUYxRjhGMUYxRjFGMUY4RjpGN0Y6LUY8NiRGOEY+RjFGOiosRjBGMUYzRjFGQEYxRjZGOi1GPDYkRjdGPkYxRjEqKkYwRjFGM0YxRjdGOi0tJkkiREdGJTYjRjE2I0Y8Rj1GMUY6KiYjRjEiIiRGMUY4RjFGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMkdGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwoKiYpSSJZR0YoRjJGMUkieEdGKCEiIkYxKiZGMkYxRjdGMUY5KiRGOEY5RjFGMUY4RjFGNkY5LUkiTUdGKDYkIiIhRj9GMUY5KjJGMEYxRjNGMSwoKiRGNkYxRjFGN0YxRjFGMUYxLCZGO0YxRjFGOUYxLCZGO0YxRjFGMUYxRjhGMUY3RjktRj02JEY/RjdGMUYxKixGM0YxLCgqJClGOEYyRjlGMUY7RjFGMUYxRjFGOEYxRjdGOS1GPTYkRjtGP0YxRjkqLEYwRjFGM0YxRkFGMUY2RjktRj02JEY3Rj9GMUYxKipGMEYxRjNGMUY3RjktLSZJIkRHRiU2I0YxNiNGPUY+RjFGOSomI0YxIiIkRjFGOEY5RjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmM0dGKCwuKi4jIiIiIiIjRjFJInRHRihGMSwoKiZJIllHRihGMSlJInhHRihGMkYxRjFGNkYxKiZGMkYxRjhGMSEiIkYxRjZGOkY3RjotSSJNR0YoNiQiIiFGPkYxRjoqMkYwRjFGM0YxLCgqJEY3RjFGMUY4RjFGMUYxRjEsJkY2RjFGMUY6RjEsJkY2RjFGMUYxRjFGNkY6RjhGOi1GPDYkRj5GOEYxRjEqLEYzRjEsKCokKUY2RjJGMUYxRjZGMUYxRjFGMUY2RjpGOEY6LUY8NiRGNkY+RjFGOiosRjBGMUYzRjFGQEYxRjdGOi1GPDYkRjhGPkYxRjEqKkYwRjFGM0YxRjhGOi0tJkkiREdGJTYjRjE2I0Y8Rj1GMUY6KiYjRjEiIiRGMUY2RjFGMTcjRi4=

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

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+SSZmM2Jpc0dGKCwuKigsKCooIiIjIiIiSSJZR0YoRjNJInhHRihGM0YzKiQpRjVGMkYzISIiRjNGOEYzRjVGOC1JIk1HRig2JCIiIUY8RjNGOCosLChGNkYzRjVGM0YzRjNGMywmRjRGM0YzRjhGMywmRjRGM0YzRjNGM0Y0RjgtRjo2JEY8RjVGM0YzKihGPkYzRjVGOC1GOjYkRjVGPEYzRjgqKkYyRjNGPkYzRjVGOC1GOjYkKiRGNUY4RjxGM0Y4KiYiIiRGMy0tJkkiREdGJTYjRjM2I0Y6RjtGM0Y4KiwjRjJGS0YzRjRGMywmKiZGMkYzRjdGM0YzRjNGM0YzSSJ0R0YoRjhGNUY4RjM3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZmNGJpc0dGKCwuKiYsKCooIiIjIiIiSSJZR0YoRjNJInhHRihGMyEiIiokKUY1RjJGM0YzRjNGM0YzLUkiTUdGKDYkIiIhRjxGM0YzKi4sKEY3RjNGNUYzRjNGM0YzLCZGNEYzRjNGNkYzLCZGNEYzRjNGM0YzRjRGNkY1RjYtRjo2JEY8KiRGNUY2RjNGMyomLCgqJkYyRjNGOEYzRjYqJkYyRjNGNUYzRjZGMkY2RjMtRjo2JEY1RjxGM0YzKiYsKEY3RjZGNUY2RjNGNkYzLUY6NiRGQ0Y8RjNGMyooIiIkRjMtLSZJIkRHRiU2I0YzNiNGOkY7RjNGNUYzRjYqKiNGMkZPRjNGNEYzLCZGN0YzRjJGM0YzSSJ0R0YoRjZGMzcjRi4=

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+SSNmOEdGKCwuKigsKCooIiIjIiIiSSJZR0YoRjNJInhHRihGM0YzKiQpRjVGMkYzISIiRjNGOEYzRjVGOC1JIk1HRig2JCIiIUY8RjNGOCosLChGNkYzRjVGM0YzRjNGMywmRjRGM0YzRjhGMywmRjRGM0YzRjNGM0Y0RjgtRjo2JEY8RjVGM0Y4KjBGMkYzRjdGOEY+RjNGP0YzRkBGM0Y0RjgtRjo2JEY8KiRGNUY4RjNGMyoqIiIkRjNGPkYzRjVGOC1GOjYkRjVGPEYzRjgqJkZIRjMtLSZJIkRHRiU2I0YzNiNGOkY7RjNGOCoqRjJGM0Y0RjNGNUY4SSJ0R0YoRjhGMzcjRi4=

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=

let us check with the paper f8pap:=(x+1+1/x)*(Y-1/Y)*(x*M(0,x)-2/x*M(0,1/x))+3*(x+1/x+1)*M(x,0)-2/x/t*Y+3*(D[1](M))(0, 0)+M(0,0)*(2*Y-x-1/x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZmOHBhcEdGKCwsKigsKEkieEdGKCIiIiokRjEhIiJGMkYyRjJGMiwmSSJZR0YoRjIqJEY2RjRGNEYyLCYqJkYxRjItSSJNR0YoNiQiIiFGMUYyRjIqKCIiI0YyRjFGNC1GOzYkRj1GM0YyRjRGMkYyKigiIiRGMkYwRjItRjs2JEYxRj1GMkYyKipGP0YyRjZGMkYxRjRJInRHRihGNEY0KiZGQ0YyLS0mSSJER0YlNiNGMjYjRjs2JEY9Rj1GMkYyKiYtRjtGT0YyLCgqJkY/RjJGNkYyRjJGMUY0RjNGNEYyRjI3I0Yu

factor(f8+f8pap);

IiIh

What we should remember from the above subsection f8:=-(2*Y*x-x^2-1)*M(0, 0)/x-(x^2+x+1)*(Y-1)*(Y+1)*M(0, x)/Y+(2*(x^2+x+1))*(Y-1)*(Y+1)*M(0, 1/x)/(x^2*Y)-(3*(x^2+x+1))*M(x, 0)/x-3*(D[1](M))(0, 0)+2*Y/(x*t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmOEdGKCwuKigsKCooIiIjIiIiSSJZR0YoRjNJInhHRihGM0YzKiQpRjVGMkYzISIiRjNGOEYzLUkiTUdGKDYkIiIhRjxGM0Y1RjhGOCosLChGNkYzRjVGM0YzRjNGMywmRjRGM0YzRjhGMywmRjRGM0YzRjNGMy1GOjYkRjxGNUYzRjRGOEY4KjBGMkYzRj5GM0Y/RjNGQEYzLUY6NiRGPCokRjVGOEYzRjdGOEY0RjhGMyoqIiIkRjNGPkYzLUY6NiRGNUY8RjNGNUY4RjgqJkZIRjMtLSZJIkRHRiU2I0YzNiNGOkY7RjNGOCoqRjJGM0Y0RjNJInRHRihGOEY1RjhGMzcjRi4=

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

KytJInRHNiIsJComIyIiIiIiJEYoSSJ4R0YkRihGKCIiIywmRihGKComRidGKClGKkYrRihGKEYpLCpGK0YoKiZGKUYoRipGKEYoKiYjRitGKUYoRi5GKEYoKiZGMkYoKUYqRilGKEYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiY=

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

KytJInRHNiIqJkkieEdGJCIiIiwmRiZGJ0YnRidGJyIiIyomRiZGJywoKiZGKUYnKUYmRilGJ0YnKiZGKUYnRiZGJ0YnRilGJ0YnIiIkKiZGJkYnLCoqJiIiJ0YnKUYmRi9GJ0YnKiYiIzdGJ0YtRidGJyomIiNCRidGJkYnRiciIzxGJ0YnIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIiY=

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

KiYpSSJ0RzYiIiIjIiIiLC4qJiIlTUtGJylGJCIiJ0YnRicqJiIkTyZGJylGJCIiJkYnRicqJiIjISlGJylGJCIiJUYnRicqJiIjPEYnKUYkIiIkRidGJyomRiZGJ0YjRidGJ0YkRidGJw==

KiZJInRHNiIiIiIsLiomIiVNS0YlKUYjIiInRiVGJSomIiRPJkYlKUYjIiImRiVGJSomIiMhKUYlKUYjIiIlRiVGJSomIiM8RiUpRiMiIiRGJUYlKiYiIiNGJSlGI0Y4RiVGJUYjRiVGJQ==

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-(2*Y*x-x^2-1)*M00/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdmOGZvcm1HRigsLCooSSJ0R0YoISIiKUkjRGVHRigjIiIiIiIjRjUsJiomSSJ4R0YoRjUtSSJNR0YoNiQiIiFGOUY1RjUqKEY2RjVGOUYxLUY7NiRGPSokRjlGMUY1RjFGNUY1KigiIiRGNSwoRjlGNUZBRjVGNUY1RjUtRjs2JEY5Rj1GNUYxKipGNkY1SSJZR0YoRjVGMEYxRjlGMUY1KiZGQ0Y1SSRNMTBHRihGNUYxKigsKCooRjZGNUZIRjVGOUY1RjUqJClGOUY2RjVGMUY1RjFGNUkkTTAwR0YoRjVGOUYxRjE3I0Yu

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);

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

Eq.(45) of the paper - at the moment eqRSpap:=sqrt(De)*(S(x)-2*S(1/x)+(x*R0-t)/t/(1+x+x^2))-3*(x+1+1/x)*R(x)-3*R1-(1-t*(x+1/x))/t/(1+x+x^2)*(x*R0-t)+(x+1/x)*R0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShlcVJTcGFwR0YoLCwqJilJI0RlR0YoIyIiIiIiI0YzLCgtSSJTR0YoNiNJInhHRihGMyomRjRGMy1GNzYjKiRGOSEiIkYzRj4qKCwmKiZJI1IwR0YoRjNGOUYzRjNJInRHRihGPkYzRkNGPiwoKiQpRjlGNEYzRjNGOUYzRjNGM0Y+RjNGM0YzKigiIiRGMywoRjlGM0Y9RjNGM0YzRjMtSSJSR0YoRjhGM0Y+KiZGSEYzSSNSMUdGKEYzRj4qKiwmRjNGMyomRkNGMywmRjlGM0Y9RjNGM0Y+RjNGQ0Y+RkRGPkZARjNGPiomRlFGM0ZCRjNGMzcjRi4=

factor(eqRS-eqRSpap);

IiIh

The series Sh and Rh. Using the new series Sh we observe a nice simplifcation on the left-hand side. Rh is the right-hand side of the equation. collect(subs(S(x)=Sh(x)+(3*R0/t-2*x-1/x)/(1+x+1/x)/3, S(1/x)=Sh(1/x)+(3*R0/t-2/x-x)/(1+x+1/x)/3, eqRS),[De,Sh,R,R1,R0],factor);

LCwqJiwmLUkjU2hHNiI2I0kieEdGJyIiIiomIiIjRiotRiY2IyokRikhIiJGKkYwRiopSSNEZUdGJyNGKkYsRipGKioqIiIkRiosKCokKUYpRixGKkYqRilGKkYqRipGKkYpRjAtSSJSR0YnRihGKkYwKiZGNUYqSSNSMUdGJ0YqRjAqLEYpRjAsLiomSSJ0R0YnRiopRikiIiVGKkYqKihGLEYqRkBGKilGKUY1RipGKiooRixGKkZARipGOEYqRioqKEYsRipGQEYqRilGKkYqRjdGMEZARipGKkZARjBGNkYwSSNSMEdGJ0YqRioqKCwoKiZGQEYqRjhGKkYqRkBGKkYpRjBGKkYpRjBGNkYwRjA=

ShS:=S(x)-(3*R0/t-2*x-1/x)/(1+x+1/x)/3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTaFNHRigsJi1JIlNHRig2I0kieEdGKCIiIiooI0YzIiIkRjMsKCooRjZGM0kjUjBHRihGM0kidEdGKCEiIkYzKiYiIiNGM0YyRjNGOyokRjJGO0Y7RjMsKEYyRjNGPkYzRjNGM0Y7Rjs3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSaFJHRigsKioqIiIkIiIiLCgqJClJInhHRigiIiNGMUYxRjVGMUYxRjFGMUY1ISIiLUkiUkdGKDYjRjVGMUYxKiZGMEYxSSNSMUdGKEYxRjEqLEY1RjcsLiomSSJ0R0YoRjEpRjUiIiVGMUYxKihGNkYxRkBGMSlGNUYwRjFGMSooRjZGMUZARjFGNEYxRjEqKEY2RjFGQEYxRjVGMUYxRjNGN0ZARjFGMUZARjdGMkY3SSNSMEdGKEYxRjcqKCwoKiZGQEYxRjRGMUYxRkBGMUY1RjdGMUY1RjdGMkY3RjE3I0Yu

The values of S(jj) and S(1/jj). What is denoted jj in this session is the cubic root of unity denoted zeta in the paper jj:=exp(2*I*Pi/3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNqakdGKCwmIyEiIiIiIyIiIiomXiMjRjJGMUYyIiIkRjVGMjcjRi4=

in the sequel 1/jj will be often represented by this expressionsimplify(1/jj);

LCQqJCwmKiZeIyIiIkYnIiIkI0YnIiIjRichIiJGJ0YrRio=

We expand the equation at x=jj op(1,simplify(factor(series(subs(De=Delta,eqRS),x=jj,3)),symbolic));

KioqJiIiJSIiIl4jRiVGJUYlLCZGJkYlKiQpIiIkI0YlIiIjRiUhIiJGJSwqKiYsJkkidEc2IkYlRiVGJUYlLUkiU0dGMjYjLCQqJkYsRiUsJiomRiZGJUYpRiVGJUYlRi1GLUYlRiVGJSomLCYqJkYrRiVGMUYlRi1GK0YtRiUtRjQ2IywmRitGLSomKiZGK0YlRiZGJUYlRilGJUYlRiVGJSomRitGJUkjUjBHRjJGJUYlKiYjRipGLEYlSSNSMUdGMkYlRiVGJSksJkYoRiVGJkYlRixGLQ==

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

LCoqJiwmSSJ0RzYiIiIiRidGJ0YnLUkiU0dGJjYjLCQqJiIiI0YnLCYqJl4jRidGJykiIiQjRidGLUYnRidGJyEiIkY0RidGJ0YnKiYsJiomRi1GJ0YlRidGNEYtRjRGJy1GKTYjLCZGM0Y0KiYqJkYzRidGMEYnRidGMUYnRidGJ0YnSSNSMEdGJkY0KiZGMkYnSSNSMUdGJkYnRjQ=

This gives the following relations between S(jj) and S(1/jj) and proves that they are equal solve({%%,%},{S(2/(I*sqrt(3)-1)),S(-1/2+I*sqrt(3)*(1/2))});

PCQvLUkiU0c2IjYjLCQqJiIiIyIiIiwmKiZeI0YrRispIiIkI0YrRipGK0YrRishIiJGMkYrLCQqJiwmSSNSMEdGJkYrKiZGMEYrSSNSMUdGJkYrRitGKywmSSJ0R0YmRitGK0YrRjJGMi8tRiU2IywmRjFGMiomKiZGMUYrRi5GK0YrRi9GK0YrRjM=

Equation (48) in the paper Sjsol:=-(R0+3*R1)/(t+1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTanNvbEdGKCwkKiYsJkkjUjBHRigiIiIqJiIiJEYySSNSMUdGKEYyRjJGMiwmSSJ0R0YoRjJGMkYyISIiRjg3I0Yu

map(simplify,series(subs(x=jj,Sex(7)),t,7))

Ky1JInRHNiIhIiIiIiMhIzYiIiUhI0kiIiYhJGkjIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiIo

R0ex:=proc(n) coeff(Rex(n),x,0): end: R1ex:=proc(n) coeff(Rex(n),x,1): end: Let's check this new expression R1ex(n),Sjsol),t,n));

KzNJInRHNiIhIiIiIiMhIzYiIiUhI0kiIiYhJGkjIiInISVlOCIiKCElJ1sqIiIpISZxJ2YiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

eqRS;

LCwqJiwqLUkiU0c2IjYjSSJ4R0YnIiIiKiYiIiNGKi1GJjYjKiRGKSEiIkYqRjAqKkYpRipJInRHRidGMCwoKiQpRilGLEYqRipGKUYqRipGKkYwSSNSMEdGJ0YqRioqJEYzRjBGMEYqKUkjRGVHRicjRipGLEYqRioqKiIiJEYqRjNGKkYpRjAtSSJSR0YnRihGKkYwKiZGPEYqSSNSMUdGJ0YqRjAqLEYpRjAsLiomRjJGKilGKSIiJUYqRioqKEYsRipGMkYqKUYpRjxGKkYqKihGLEYqRjJGKkY1RipGKiooRixGKkYyRipGKUYqRipGNEYwRjJGKkYqRjJGMEYzRjBGNkYqRioqKCwoKiZGMkYqRjVGKkYqRjJGKkYpRjBGKkYpRjBGM0YwRjA=

Partial fraction expansions factor(expand(rationalize(1/(1-jj)/(1-jj*x)+1/(1-1/jj)/(1-x/jj))))

KiQsKCokKUkieEc2IiIiIyIiIkYpRiZGKUYpRikhIiI=

factor(expand(rationalize( 2/3*(1/(1-jj)/(1-jj*x)+1/(1-1/jj)/(1-x/jj)) +1/(1-jj)^2/(1-jj*x)^2+1/(1-1/jj)^2/(1-x/jj)^2)))

KiQpLCgqJClJInhHNiIiIiMiIiJGKkYnRipGKkYqRikhIiI=

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

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

The square of the RHS RHS2:=subs(De=0,eqRS)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVSSFMyR0YoKiQpLCoqKiIiJCIiIiwoKiQpSSJ4R0YoIiIjRjNGM0Y3RjNGM0YzRjNGNyEiIi1JIlJHRig2I0Y3RjNGOSomRjJGM0kjUjFHRihGM0Y5KixGN0Y5LC4qJkkidEdGKEYzKUY3IiIlRjNGMyooRjhGM0ZCRjMpRjdGMkYzRjMqKEY4RjNGQkYzRjZGM0YzKihGOEYzRkJGM0Y3RjNGM0Y1RjlGQkYzRjNGQkY5RjRGOUkjUjBHRihGM0YzKigsKComRkJGM0Y2RjNGM0ZCRjNGN0Y5RjNGN0Y5RjRGOUY5RjhGMzcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShuZWdSSFMyR0YoLCYqJiksJiomIiIjIiIiSSNSMEdGKEY0RjRJInRHRihGNEYzRjQpSSJ4R0YoRjMhIiJGNCoqRjNGNEYxRjQsKiomIiInRjRJI1IxR0YoRjRGNEYyRjRGNkY5RjRGOUY0RjhGOUY0NyNGLg==

If we want to go one step further in the expansion, one needs R2. So we will not go one step further map(factor,subs(((D@@2)(R))(0)=2*R2,R(0)=R0,D(R)(0)=R1,series(RHS2,x,3)))

KytJInhHNiIqJCksJiomIiIjIiIiSSNSMEdGJEYqRipJInRHRiRGKkYpRiohIiMsJCooRilGKkYnRiosKiomIiInRipJI1IxR0YkRipGKkYoRipGLCEiIkYqRjRGKkYqRjQqJiw8KigiIztGKilGK0YpRipGLEYqRioqKiIjT0YqRitGKkYzRipGLEYqRioqKiIjN0YqRitGKkkjUjJHRiRGKkYsRipGKiooRjJGKkYrRiopRixGKUYqRioqKEY7RiopRjNGKUYqRixGKkYqKihGMkYqRjNGKkZARipGNCooRjJGKkY+RipGQEYqRioqJiIiJEYqKUYsRkZGKkYqKiYiIiVGKkY5RipGKiooRilGKkYrRipGLEYqRioqKEY9RipGM0YqRixGKkY0KiZGSUYqRkBGKkYqRixGKkYqRixGNCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRio=

The square of the left-hand side LHS2:=subs(De=Delta,op(1,eqRS)^2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVMSFMyR0YoKiYpLCotSSJTR0YoNiNJInhHRigiIiIqJiIiI0Y1LUYyNiMqJEY0ISIiRjVGOyoqRjRGNUkidEdGKEY7LCgqJClGNEY3RjVGNUY0RjVGNUY1RjtJI1IwR0YoRjVGNSokRj5GO0Y7RjdGNSwmKiQpLCZGNUY1KiZGPUY1LCZGNEY1RjpGNUY1RjtGN0Y1RjUqKCIiJUY1KUY9RjdGNSksKEY0RjVGOkY1RjVGNUY3RjVGO0Y1NyNGLg==

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

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

IiIn

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxMSFMyc3BlY2lhbEdGKCwkKjIiIiUiIiIsJiomSSNSMEdGKEYxSSJ4R0YoRjFGMUkidEdGKCEiIkYxLCoqJkY2RjEpRjUiIiNGMUYxKihGO0YxRjZGMUY1RjFGMUY2RjFGNUYxRjEsKiooIiIkRjFGNkYxRjpGMUYxRjxGMSomRj9GMUY2RjFGMUY1RjdGMUY2RjcsKCokRjpGMUYxRjVGMUYxRjFGN0Y6RjctSSJTR0YoNiMqJEY1RjdGMUYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpMSFMybWl4ZWRHRigsJCouIiIlIiIiLCoqJkkidEdGKEYxKUkieEdGKCIiI0YxRjEqKEY3RjFGNEYxRjZGMUYxRjRGMUY2RjFGMSwqKigiIiRGMUY0RjFGNUYxRjFGOEYxKiZGO0YxRjRGMUYxRjYhIiJGMUY1Rj0tSSJTR0YoNiNGNkYxLUY/NiMqJEY2Rj1GMUYxNyNGLg==

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

LCgqKCIjNyIiIilJI1IwRzYiIiIjRiUpSSJ4R0YoRilGJUYlKixGKUYlRidGJUkidEdGKCEiIiwqKigiIiRGJSlGLUYpRiUtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiRJIkRHRjdGKTYjSSJTR0YoNiMiIiFGJUYlKigiIztGJUYnRiVGLUYlRiUqJiIiJ0YlRjJGJUYuKiYiIiVGJUYnRiVGJUYlRitGJUYlLUkiT0dGNzYjRiVGJQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5wb3NudW1zcGVjaWFsR0YoLCQqKCIjNyIiIilJI1IwR0YoIiIjRjEpSSJ4R0YoRjRGMUYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5uZWdudW1zcGVjaWFsR0YoLCYqMCIiJSIiIiwmKiZJI1IwR0YoRjFJInhHRihGMUYxSSJ0R0YoISIiRjEsKiomRjZGMSlGNSIiI0YxRjEqKEY7RjFGNkYxRjVGMUYxRjZGMUY1RjFGMSwqKigiIiRGMUY2RjFGOkYxRjFGPEYxKiZGP0YxRjZGMUYxRjVGN0YxRjZGN0Y6RjctSSJTR0YoNiMqJEY1RjdGMUYxKigiIzdGMSlGNEY7RjFGOkYxRjc3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZGZXhwckdGKCwmKjAiIiUiIiIsJiomSSJ0R0YoRjFJInhHRihGMSEiIkkjUjBHRihGMUYxLCoqJkY0RjEpRjUiIiNGMUYxKihGO0YxRjRGMUY1RjFGMUY0RjFGNUYxRjEsKiooIiIkRjFGNEYxRjpGMUYxRjxGMSomRj9GMUY0RjFGMUY1RjZGMSlGNUY/RjYtSSJTR0YoNiNGNUYxRjRGNkYxKigiIzdGMSlGN0Y7RjFGOkY2RjY3I0Yu

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

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

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

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

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5ub25uZWdzcGVjaWFsR0YoLCQqKiIiJSIiIiw0KipGMEYxKUkjUjBHRigiIiNGMUkidEdGKEYxSSJ4R0YoRjFGMSosIiIkRjFGNUYxSSNSMUdGKEYxRjdGMUY4RjFGMSooRjpGMUY0RjFGN0YxRjEqJkY0RjFGOEYxRjEqKkY6RjFGNUYxRjtGMUY4RjFGMSomRjVGMSlGN0Y2RjEhIiIqKEY6RjFGO0YxRkBGMUZBKiZGNUYxRjdGMUZBKihGOkYxRjtGMUY3RjFGQUYxRjdGQSwoKiQpRjhGNkYxRjFGOEYxRjFGMUZBRjE3I0Yu

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

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

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);

LCQqKiIiJSIiIiw2KioiIiRGJSlJI1IwRzYiIiIjRiVJInRHRitGJSlJInhHRitGLEYlRiUqKkYkRiVGKUYlRi1GJUYvRiVGJSosRihGJUYqRiVJI1IxR0YrRiVGLUYlRi9GJUYlKihGKEYlRilGJUYtRiVGJSomRilGJUYvRiVGJSoqRihGJUYqRiVGMkYlRi9GJUYlKiZGKkYlKUYtRixGJSEiIiooRihGJUYyRiVGN0YlRjgqJkYqRiVGLUYlRjgqKEYoRiVGMkYlRi1GJUY4RiVGLUY4LCgqJEYuRiVGJUYvRiVGJUYlRjhGJQ==

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, where NEG stands for the negative part of Delta(x) S(x) S(1/x) op(2,LHS2);factor(%/Delta);

LCQqLiIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiVGKkYyLUkiU0dGKTYjRitGJS1GNDYjKiRGK0YyRiVGJQ==

LCQqKCIiJSIiIi1JIlNHNiI2I0kieEdGKEYlLUYnNiMqJEYqISIiRiVGLg==

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

LCQqMiIiIyIiIiwmKiZJI1IwRzYiRiVJInhHRilGJUYlSSJ0R0YpISIiRiUsKiomRitGJSlGKkYkRiVGJSooRiRGJUYrRiVGKkYlRiVGK0YlRipGJUYlLCoqKCIiJEYlRitGJUYvRiVGJUYwRiUqJkYzRiVGK0YlRiVGKkYsRiVGK0YsLCgqJEYvRiVGJUYqRiVGJUYlRixGL0YsLUkiU0dGKTYjRipGJUYs

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

KydJInhHNiIsJCooIiInIiIiSSNSMEdGJEYoSSJ0R0YkRihGKCEiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIh

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJiomIiIlIiIiSSRORUdHRihGMSEiIioqIiInRjFJI1IwR0YoRjFJInRHRihGMUkieEdGKEYzRjE3I0Yu

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+SSRyZXNHRigsKiomIiIlIiIiSSRORUdHRihGMSEiIioqIiInRjFJI1IwR0YoRjFJInRHRihGMUkieEdGKEYzRjEqLEYwRjEsKiomRjdGMSlGOCIiI0YxRjEqKEY9RjFGN0YxRjhGMUYxRjdGMUY4RjFGMSwqKigiIiRGMUY3RjFGPEYxRjFGPkYxKiZGQUYxRjdGMUYxRjhGM0YxRjxGMyktSSJTR0YoNiMqJEY4RjNGPUYxRjMqJiIjN0YxKUY2Rj1GMUYxNyNGLg==

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

LCQqMiIiJSIiIiwmKiZJI1IwRzYiRiVJInhHRilGJUYlSSJ0R0YpISIiRiUsKiomRitGJSlGKiIiI0YlRiUqKEYwRiVGK0YlRipGJUYlRitGJUYqRiVGJSwqKigiIiRGJUYrRiVGL0YlRiVGMUYlKiZGNEYlRitGJUYlRipGLEYlRitGLCwoKiRGL0YlRiVGKkYlRiVGJUYsRi9GLC1JIlNHRik2IyokRipGLEYlRiU=

LCQqMiIiJSIiIiwmKiZJI1IwRzYiRiVJInhHRilGJUYlSSJ0R0YpISIiRiUsKiomRitGJSlGKiIiI0YlRiUqKEYwRiVGK0YlRipGJUYlRitGJUYqRiVGJSwqKigiIiRGJUYrRiVGL0YlRiVGMUYlKiZGNEYlRitGJUYlRipGLEYlRitGLCwoKiRGL0YlRiVGKkYlRiVGJUYsRi9GLC1JIlNHRik2IyokRipGLEYlRiU=

We have already determined its negative part res:=res+negspecial;

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

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

LCQqLiksJiomSSNSMEc2IiIiIkkieEdGKEYpRilJInRHRighIiIiIiNGKSwqKiZGK0YpKUYqRi1GKUYpKihGLUYpRitGKUYqRilGKUYrRilGKkYpRiksKiooIiIkRilGK0YpRjBGKUYpRjFGKSomRjRGKUYrRilGKUYqRixGKSlGK0YtRiwpLCgqJEYwRilGKUYqRilGKUYpRi1GLEYwRixGLA==

KylJInhHNiIsJComIiIkIiIiKUkidEdGJCIiI0YoISIiISIjLCQqKEYrRihGKkYoLChGKkYsRihGLComRidGKEkjUjBHRiRGKEYoRihGKEYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiE=

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

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

In conclusion, extracting the negative part in the square of the equation 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));

Ky9JInRHNiIqJEkieEdGJCEiIiIiJSwkKiYiIiMiIiJGJkYnRiwiIiYqJiwoKiYiIzZGLClGJkYrRixGLComRihGLEYmRixGJ0YsRidGLClGJiIiJEYnIiInLCQqKEYrRiwsKiomIiNERixGNEYsRiwqJiIjN0YsRjJGLEYnKiZGLUYsRiZGLEYnRixGJ0YsKUYmRihGJ0YsIiIoKiYsLComIiRaI0YsRj9GLEYsKiYiJEojRixGNEYsRicqJiIkKyJGLEYyRixGJyomIiNHRixGJkYsRidGLUYnRiwpRiZGLUYnIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiwiIio=

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

Ky9JInRHNiIsJkkieEdGJCIiIiIiI0YnIiIlLCYqJkYoRidGJkYnRidGKUYnIiImLCoqJClGJiIiJEYnISIiKiZGKUYnKUYmRihGJ0YxKiYiIzZGJ0YmRidGJyIjS0YnIiInLCQqKEYoRidGJUYnLCpGLkYnKiZGMEYnRjNGJ0YnKiZGN0YnRiZGJ0YnIiNQRjFGJ0YxIiIoLC4qJkYsRicpRiZGLEYnRjEqJiIjR0YnKUYmRilGJ0YxKiYiJCsiRidGL0YnRjEqJiIkSiNGJ0YzRidGMSomIiRaI0YnRiZGJ0YnIiRxKkYnIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIio=

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, we find for P0 a series that is independent of x... 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));

KzFJInRHNiJJI1AwR0YkIiIhISIjIiIlISIlIiImISNLIiInISRbIiIiKCEkcSoiIiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

We can express P0 in terms of R and S by repeating the above procedure, extracting the constant term (CT) rather than the negative part resCT:=0:RHS2;

KiQpLCoqKiIiJCIiIiwoKiQpSSJ4RzYiIiIjRidGJ0YrRidGJ0YnRidGKyEiIi1JIlJHRiw2I0YrRidGLiomRiZGJ0kjUjFHRixGJ0YuKixGK0YuLC4qJkkidEdGLEYnKUYrIiIlRidGJyooRi1GJ0Y3RicpRitGJkYnRicqKEYtRidGN0YnRipGJ0YnKihGLUYnRjdGJ0YrRidGJ0YpRi5GN0YnRidGN0YuRihGLkkjUjBHRixGJ0YnKigsKComRjdGJ0YqRidGJ0Y3RidGK0YuRidGK0YuRihGLkYuRi1GJw==

map(factor,subs(((D@@2)(R))(0)=2*R2,R(0)=R0,D(R)(0)=R1,series(RHS2,x,3)));

KytJInhHNiIqJCksJiomIiIjIiIiSSNSMEdGJEYqRipJInRHRiRGKkYpRiohIiMsJCooRilGKkYnRiosKiomIiInRipJI1IxR0YkRipGKkYoRipGKiEiIkYsRjRGKkYqRjQqJiw8KigiIztGKilGK0YpRipGLEYqRioqKiIjT0YqRitGKkYzRipGLEYqRioqKiIjN0YqRitGKkkjUjJHRiRGKkYsRipGKiooRjJGKkYrRiopRixGKUYqRioqKEY7RiopRjNGKUYqRixGKkYqKihGMkYqRjNGKkZARipGNCooRjJGKkY+RipGQEYqRioqJiIiJEYqKUYsRkZGKkYqKiYiIiVGKkY5RipGKiooRilGKkYrRipGLEYqRioqKEY9RipGM0YqRixGKkY0KiZGSUYqRkBGKkYqRixGKkYqRixGNCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRio=

resCT:=resCT-coeff(%,x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZXNDVEdGKCwkKiYsPCooIiM7IiIiKUkjUjBHRigiIiNGM0kidEdGKEYzRjMqKiIjT0YzRjVGM0kjUjFHRihGM0Y3RjNGMyoqIiM3RjNGNUYzSSNSMkdGKEYzRjdGM0YzKigiIidGM0Y1RjMpRjdGNkYzRjMqKEY5RjMpRjpGNkYzRjdGM0YzKihGP0YzRjpGM0ZARjMhIiIqKEY/RjNGPUYzRkBGM0YzKiYiIiRGMylGN0ZHRjNGMyomIiIlRjNGNEYzRjMqKEY2RjNGNUYzRjdGM0YzKihGPEYzRjpGM0Y3RjNGRComRkpGM0ZARjNGM0Y3RjNGM0Y3RkRGRDcjRi4=

subs(S(0)=0,(D(S))(0)=R0/t,series(op(1,LHS2),x,3));

KydJInhHNiIsJComIiIkIiIiKUkjUjBHRiQiIiNGKCEiIiIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZXNDVEdGKCwmKiYsPCooIiM7IiIiKUkjUjBHRigiIiNGM0kidEdGKEYzRjMqKiIjT0YzRjVGM0kjUjFHRihGM0Y3RjNGMyoqIiM3RjNGNUYzSSNSMkdGKEYzRjdGM0YzKigiIidGM0Y1RjMpRjdGNkYzRjMqKEY5RjMpRjpGNkYzRjdGM0YzKihGP0YzRjpGM0ZARjMhIiIqKEY/RjNGPUYzRkBGM0YzKiYiIiRGMylGN0ZHRjNGMyomIiIlRjNGNEYzRjMqKEY2RjNGNUYzRjdGM0YzKihGPEYzRjpGM0Y3RjNGRComRkpGM0ZARjNGM0Y3RjNGM0Y3RkRGRComRkdGM0Y0RjNGRDcjRi4=

op(2,LHS2);

LCQqLiIiJSIiIiwqKiZJInRHNiJGJSlJInhHRikiIiNGJUYlKihGLEYlRihGJUYrRiVGJUYoRiVGK0YlRiUsKiooIiIkRiVGKEYlRipGJUYlRi1GJSomRjBGJUYoRiVGJUYrISIiRiUtSSJTR0YpNiNGK0YlLUY0NiMqJEYrRjJGJUYqRjJGJQ==

resCT:=resCT-4*CT;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZXNDVEdGKCwoKiYsPCooIiM7IiIiKUkjUjBHRigiIiNGM0kidEdGKEYzRjMqKiIjT0YzRjVGM0kjUjFHRihGM0Y3RjNGMyoqIiM3RjNGNUYzSSNSMkdGKEYzRjdGM0YzKigiIidGM0Y1RjMpRjdGNkYzRjMqKEY5RjMpRjpGNkYzRjdGM0YzKihGP0YzRjpGM0ZARjMhIiIqKEY/RjNGPUYzRkBGM0YzKiYiIiRGMylGN0ZHRjNGMyomIiIlRjNGNEYzRjMqKEY2RjNGNUYzRjdGM0YzKihGPEYzRjpGM0Y3RjNGRComRkpGM0ZARjNGM0Y3RjNGM0Y3RkRGRComRkdGM0Y0RjNGRComRkpGM0kjQ1RHRihGM0ZENyNGLg==

factor(subs(((D@@2)(S))(0)=2*S2,S(0)=0,(D(S))(0)=R0/t,series(op(3,LHS2),x,3)));

KylJInhHNiIsJCooIiInIiIiSSNSMEdGJEYoSSJ0R0YkRihGKCEiIiwqKihGJ0YoSSNTMkdGJEYoKUYqIiIjRihGKComRidGKClGKUYwRihGKyooIiM1RihGKUYoRipGKEYoKiYiIiVGKEYpRihGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

resCT:=resCT+coeff(%,x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZXNDVEdGKCwuKiYsPCooIiM7IiIiKUkjUjBHRigiIiNGM0kidEdGKEYzRjMqKiIjT0YzRjVGM0kjUjFHRihGM0Y3RjNGMyoqIiM3RjNGNUYzSSNSMkdGKEYzRjdGM0YzKigiIidGM0Y1RjMpRjdGNkYzRjMqKEY5RjMpRjpGNkYzRjdGM0YzKihGP0YzRjpGM0ZARjMhIiIqKEY/RjNGPUYzRkBGM0YzKiYiIiRGMylGN0ZHRjNGMyomIiIlRjNGNEYzRjMqKEY2RjNGNUYzRjdGM0YzKihGPEYzRjpGM0Y3RjNGRComRkpGM0ZARjNGM0Y3RjNGM0Y3RkRGRComIiIqRjNGNEYzRkQqJkZKRjNJI0NUR0YoRjNGRCooRj9GM0ZARjNJI1MyR0YoRjNGMyooIiM1RjNGNUYzRjdGM0YzKiZGSkYzRjVGM0YzNyNGLg==

map(factor,subs(((D@@2)(S))(0)=2*S2,S(0)=0,(D(S))(0)=R0/t,series(op(4,LHS2),x=infinity)));

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

op(1,%);

LCQqJiIjNyIiIilJI1IwRzYiIiIjRiUhIiI=

resCT:=resCT+%;

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

op(5,LHS2); nonnegspecial;

LCQqMiIiJSIiIiwmKiZJI1IwRzYiRiVJInhHRilGJUYlSSJ0R0YpISIiRiUsKiomRitGJSlGKiIiI0YlRiUqKEYwRiVGK0YlRipGJUYlRitGJUYqRiVGJSwqKigiIiRGJUYrRiVGL0YlRiVGMUYlKiZGNEYlRitGJUYlRipGLEYlLUkiU0dGKTYjKiRGKkYsRiVGK0YsLCgqJEYvRiVGJUYqRiVGJUYlRixGL0YsRiU=

LCQqKiIiJSIiIiw0KipGJEYlKUkjUjBHNiIiIiNGJUkidEdGKkYlSSJ4R0YqRiVGJSosIiIkRiVGKUYlSSNSMUdGKkYlRixGJUYtRiVGJSooRi9GJUYoRiVGLEYlRiUqJkYoRiVGLUYlRiUqKkYvRiVGKUYlRjBGJUYtRiVGJSomRilGJSlGLEYrRiUhIiIqKEYvRiVGMEYlRjVGJUY2KiZGKUYlRixGJUY2KihGL0YlRjBGJUYsRiVGNkYlRixGNiwoKiQpRi1GK0YlRiVGLUYlRiVGJUY2RiU=

factor(series(nonnegspecial,x,1));

KydJInhHNiIsLComIiM3IiIiKUkjUjBHRiQiIiNGKEYoKigiIiVGKEYqRihJInRHRiRGKCEiIiooRidGKEkjUjFHRiRGKEYuRihGLyomRi1GKEYqRihGLyomRidGKEYxRihGLyIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

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

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

factor(series(op(6,LHS2),x,3));

KytJInhHNiIsJComIiIkIiIiKUkidEdGJCIiI0YoISIiISIjLCQqKEYrRihGKkYoLChGKkYsRihGLComRidGKEkjUjBHRiRGKEYoRihGKEYsLC4qJkYnRigpRjJGK0YoRiwqKCIiJUYoRjJGKEYqRihGKEYmRigqJkY3RihGMkYoRigqJkY3RihGKkYoRihGKEYoIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKA==

resCT:=resCT+coeff(%,x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZXNDVEdGKCw4KiYsPCooIiM7IiIiKUkjUjBHRigiIiNGM0kidEdGKEYzRjMqKiIjT0YzRjVGM0kjUjFHRihGM0Y3RjNGMyoqIiM3RjNGNUYzSSNSMkdGKEYzRjdGM0YzKigiIidGM0Y1RjMpRjdGNkYzRjMqKEY5RjMpRjpGNkYzRjdGM0YzKihGP0YzRjpGM0ZARjMhIiIqKEY/RjNGPUYzRkBGM0YzKiYiIiRGMylGN0ZHRjNGMyomIiIlRjNGNEYzRjMqKEY2RjNGNUYzRjdGM0YzKihGPEYzRjpGM0Y3RjNGRComRkpGM0ZARjNGM0Y3RjNGM0Y3RkRGRComRjxGM0Y0RjNGRComRkpGM0kjQ1RHRihGM0ZEKihGP0YzRkBGM0kjUzJHRihGM0YzKigiIzVGM0Y1RjNGN0YzRjNGTEZEKiZGPEYzRjpGM0ZEKiZGR0YzRkBGM0YzKiZGSkYzRjVGM0YzKiZGSkYzRjdGM0YzRjNGMzcjRi4=

eqCT:=collect(resCT/4,CT,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcUNUR0YoLCZJI0NUR0YoISIiKigjIiIiIiIjRjMsNiooIiIkRjNJI1MyR0YoRjMpSSJ0R0YoRjdGM0YwKigiIzlGMylJI1IwR0YoRjRGM0Y6RjNGMyoqIiM9RjNGPkYzSSNSMUdGKEYzRjpGM0YzKioiIidGM0Y+RjNJI1IyR0YoRjNGOkYzRjMqKEY0RjNGPkYzKUY6RjRGM0YwKihGQEYzKUZBRjRGM0Y6RjNGMyooRjdGM0ZBRjNGRkYzRjMqKEY3RjNGREYzRkZGM0YzKiZGNEYzRj1GM0YzKiZGPkYzRjpGM0YwRjNGOkYwRjA3I0Yu

Let us check this equation for the CT in x of Delta S(x) S(1/x). Correct. n:=12: series(subs(CT=0,R0=coeff(Rex(n),x,0), R1=coeff(Rex(n),x,1), R2=coeff(Rex(n),x,2), S2=coeff(Sex(n),x,2), eqCT),t,n);

KzNJInRHNiIiIiMiIiVGJiIiJiIjSyIiJyIkWyIiIigiJHEqIiIpIiVnZCIiKiImRXokIiM1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM2

P0ex:=proc(n) factor(series(Delta*Sex(n)*subs(x=1/x,Sex(n)),t,n+1)): coeff(convert(%,polynom),x,0); end: P0ex(10);

LDAqJiImRXokIiIiKUkidEc2IiIjNUYlRiUqJiIlZ2RGJSlGJyIiKkYlRiUqJiIkcSpGJSlGJyIiKUYlRiUqJiIkWyJGJSlGJyIiKEYlRiUqJiIjS0YlKUYnIiInRiVGJSomIiIlRiUpRiciIiZGJUYlKiYiIiNGJSlGJ0Y7RiVGJQ==

Comparison with the paper eqSpap:=Delta*(S(x)^2+S(1/x)^2-S(x)*S(1/x)+(S(x)*(x*t-R0)+S(1/x)*(t/x-R0))/t/(1+x+1/x)) -(R0+3*R1)*((2*R0+t)*(x+1/x+(1+t)/t/(1+x+1/x))-1-t) +(1+4*t)*(x+1/x)*R0-(t^2+t*R0+R0^2)*(x^2+1/x^2)+P0;

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

factor(eqS-eqSpap);

IiIh

eqS:=Delta*(S(x)^2+S(1/x)^2-S(x)*S(1/x)+(S(x)*(x*t-R0)+S(1/x)*(t/x-R0))/t/(1+x+1/x)) -(R0+3*R1)*((2*R0+t)*(x+1/x+(1+t)/t/(1+x+1/x))-1-t) +(1+4*t)*(x+1/x)*R0-(t^2+t*R0+R0^2)*(x^2+1/x^2)+P0;

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

The constant term in Delta S(x) S(1/x) P0ex:=proc(n) factor(series(Delta*Sex(n)*subs(x=1/x,Sex(n)),t,n+1)): coeff(convert(%,polynom),x,0); end: Let us check that no simplification occurs in the terms with no S with(gfun):Deltaroots:=map(allvalues,algeqtoseries(numer(Delta),t,x,4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStEZWx0YXJvb3RzR0YoNyYrLUkidEdGKCMiIiIiIiQhIiIjISIjRjMiIiEhIiRGMiEiJyIiIy1JIk9HRiU2I0YyRjMrLUYwRjRGNEY2RjdGMkYyRjZGOkY7RjMrLUYwRjNGMiIiJ0Y6IiNSRjMiJCc9IiIlRjsiIiYrLUYwRjRGMkY6RjohIiZGMyIjOUZDRjtGRDcjRi4=

n:=10: for i from 1 to 4 do series(subs(S(x)=0,S(1/x)=0,R0=R0ex(n),R1=R1ex(n),P0=P0ex(n), x=Deltaroots[i],eqS),t) od;

Ky1JInRHNiIjISIiIiIqIiIhIyIiJUYnIiIiIyIjQUYnIiIjIyIiKUYnIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGKg==

Ky1JInRHNiIhIiIiIiEhIiUiIiIhIiMiIiMhIikiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJQ==

Ky1JInRHNiIjISIiIiIqIiIhIyIiJUYnIiIiIyIjQUYnIiIjIyIiKUYnIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGKg==

Ky1JInRHNiIhIiIiIiEhIiUiIiIhIiMiIiMhIikiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJQ==

Equation in terms of S-hat (at the moment Eq.(50) of the paper) eqSh:=collect(subs(isolate(ShS-Sh(x),S(x)), subs(x=1/x, isolate(ShS-Sh(x),S(x))), eqS/Delta),[Sh],factor)

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

Let us check that no simplification occurs in the terms with no Sh n:=15: for i from 1 to 4 do series(subs(Sh(x)=0,Sh(1/x)=0,R0=R0ex(n),R1=R1ex(n),P0=P0ex(n), x=Deltaroots[i],numer(op(4,eqSh))),t) od;

KylJInRHNiIjIiIiIiUoPSMhIiUjISM1RichIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiEiIw==

KylJInRHNiIiIiQhIiUiI1UhIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIhIiM=

KylJInRHNiIiIiQiIiUiIz0iIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

KylJInRHNiIiIiQiIiUhIiciIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

numer(factor(subs(x=1,eqS))); indets(%);

LEgqKCIjJyoiIiIpLUkiU0c2IjYjRiUiIiNGJSlJInRHRikiIiRGJSEiIiooIiM3RiVGJkYlKUYtRitGJUYvKioiI2tGJUYnRiVJI1IwR0YpRiVGMkYlRiUqKEY0RiVGJ0YlRixGJUYvKihGLkYlRiZGJUYtRiVGJSoqIiIpRiVGJ0YlRjVGJUYtRiVGJSooRjlGJUYnRiVGMkYlRi8qKCIjP0YlKUY1RitGJUYtRiVGLyoqIiNVRiVGNUYlSSNSMUdGKUYlRi1GJUYvKigiIzlGJUY1RiVGMkYlRiUqKEYxRiVGQEYlRjJGJUYvKiYiIidGJUYsRiVGLyooRitGJUYnRiVGNUYlRi8qKEYrRiVGJ0YlRi1GJUYlKihGLkYlSSNQMEdGKUYlRi1GJUYlKiZGK0YlRj1GJUYvKihGRUYlRjVGJUZARiVGLyooRjlGJUY1RiVGLUYlRiUqKEZFRiVGQEYlRi1GJUYl

PCZJI1AwRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

ShS;

LCYtSSJTRzYiNiNJInhHRiUiIiIqKCNGKCIiJEYoLCgqKEYrRihJI1IwR0YlRihJInRHRiUhIiJGKComIiIjRihGJ0YoRjAqJEYnRjBGMEYoLChGJ0YoRjNGKEYoRihGMEYw

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);

KjAsKiomSSJ0RzYiIiIiKUkieEdGJiIiI0YnRicqKEYqRidGJUYnRilGJ0YnRiVGJ0YpRidGJywqKigiIiRGJ0YlRidGKEYnRidGK0YnKiZGLkYnRiVGJ0YnRikhIiJGJywoKihGKkYnRiVGJ0YoRidGMCooRi5GJ0kjUjBHRiZGJ0YpRidGJ0YlRjBGJ0YlRjAsKCokRihGJ0YnRilGJ0YnRidGMEYoRjApLUkiU0dGJjYjRilGKkYn

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

KiwpSSJ4RzYiIiIjISIiLFx0KigpSSNSMEdGJUYmIiIiKUkidEdGJUYmRiwpRiQiIilGLEYnKihGK0YsKUYuIiIkRixGL0YsRicqKCIiJUYsKUYuRjVGLEYvRixGJyoqRjVGLEYqRixGLUYsKUYkIiIoRixGJyosIiInRixGK0YsSSNSMUdGJUYsRi1GLEY4RixGJyoqIiM1RixGK0YsRjJGLEY4RixGLCoqRjNGLEY8RixGMkYsRjhGLEYnKihGPkYsRjZGLEY4RixGJyoqIiM4RixGKkYsRi1GLClGJEY7RixGJyosIiM3RixGK0YsRjxGLEYtRixGQ0YsRicqKiIjR0YsRitGLEYyRixGQ0YsRiwqKEYrRixGLUYsRjhGLEYsKipGM0YsRjxGLEYyRixGQ0YsRicqKCIjO0YsRjZGLEZDRixGJyooRiZGLEYyRixGOEYsRicqKEkjUDBHRiVGLEYtRixGQ0YsRiwqKkZHRixGKkYsRi1GLClGJCIiJkYsRicqLCIjSUYsRitGLEY8RixGLUYsRlBGLEYnKioiI1dGLEYrRixGMkYsRlBGLEYsKioiIipGLEYrRixGLUYsRkNGLEYsKipGV0YsRjxGLEYyRixGUEYsRicqKkYzRixGPEYsRi1GLEZDRixGLCooIiM9RixGNkYsRlBGLEYnKipGJkYsRk5GLEYtRixGUEYsRiwqKiIjS0YsRipGLEYtRiwpRiRGNUYsRicqKkY7RixGKkYsRi5GLEZQRixGJyosRlNGLEYrRixGPEYsRi1GLEZpbkYsRicqLEY7RixGK0YsRjxGLEYuRixGUEYsRicqKkZVRixGK0YsRjJGLEZpbkYsRiwqKkZRRixGK0YsRi1GLEZQRixGLCoqRjtGLEY8RixGMkYsRmluRixGJyoqRjNGLEY8RixGLUYsRlBGLEYsKigiIzpGLEY2RixGaW5GLEYnKihGNUYsRjJGLEZQRixGJyomRi1GLEZDRixGLCoqRjNGLEZORixGLUYsRmluRixGLCoqRkdGLEYqRixGLUYsKUYkRjNGLEYnKipGJkYsRipGLEYuRixGaW5GLEYnKixGU0YsRitGLEY8RixGLUYsRmdvRixGJyosRjtGLEYrRixGPEYsRi5GLEZpbkYsRicqKkZTRixGK0YsRjJGLEZnb0YsRiwqKiIjOUYsRitGLEYtRixGaW5GLEYsKipGM0YsRitGLEYuRixGUEYsRicqKkZXRixGPEYsRjJGLEZnb0YsRicqKkY7RixGPEYsRi1GLEZpbkYsRiwqKEY+RixGNkYsRmdvRixGJyoqRiZGLEZORixGLUYsRmdvRixGLCoqRkJGLEYqRixGLUYsRiNGLEYnKipGO0YsRipGLEYuRixGZ29GLEYnKihGJkYsRipGLEZpbkYsRiwqLEZFRixGK0YsRjxGLEYtRixGI0YsRicqLEY7RixGK0YsRjxGLEYuRixGZ29GLEYnKipGRUYsRitGLEYyRixGI0YsRiwqKkZRRixGK0YsRi1GLEZnb0YsRiwqKkYzRixGPEYsRjJGLEYjRixGJyoqRjNGLEY8RixGLUYsRmdvRixGLCooRjtGLEY2RixGI0YsRicqKEYmRixGMkYsRmdvRixGJyomRi1GLEZpbkYsRiwqKEZORixGLUYsRiNGLEYsKipGNUYsRipGLEYtRixGJEYsRicqLEY7RixGK0YsRjxGLEYtRixGJEYsRicqKkY1RixGK0YsRjJGLEYkRixGLCoqRlFGLEYrRixGLUYsRiNGLEYsKihGK0YsRi5GLEZnb0YsRicqKkYzRixGPEYsRjJGLEYkRixGJyoqRjNGLEY8RixGLUYsRiNGLEYsKihGJkYsRjZGLEYkRixGJyomRipGLEYtRixGJyomRitGLEYyRixGJyooRitGLEYtRixGJEYsRiwqJEY2RixGJ0YsRi1GJyksKCokRiNGLEYsRiRGLEYsRixGJkYnLUkiU0dGJTYjRiRGLA==

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

KydJInhHNiIsJCooSSJ0R0YkISIiLCgqJClJI1IwR0YkIiIjIiIiRi4qJkYsRi5GJ0YuRi4qJClGJ0YtRi5GLkYuRixGLkYoRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLiIiIQ==

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. We apply the first part of the lemma op(5,eqSdec);

KjAsKiomSSJ0RzYiIiIiKUkieEdGJiIiI0YnRicqKEYqRidGJUYnRilGJ0YnRiVGJ0YpRidGJywqKigiIiRGJ0YlRidGKEYnRidGK0YnKiZGLkYnRiVGJ0YnRikhIiJGJywoRiRGMCooRi5GJ0kjUjBHRiZGJ0YpRidGJyomRipGJ0YlRidGMEYnRiVGMCwoKiRGKEYnRidGKUYnRidGJ0YwRihGMCktSSJTR0YmNiMqJEYpRjBGKkYn

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRudW1HRigqLiwqKiZJInRHRigiIiIpSSJ4R0YoIiIjRjJGMiooRjVGMkYxRjJGNEYyRjJGMUYyRjRGMkYyLCoqKCIiJEYyRjFGMkYzRjJGMkY2RjIqJkY5RjJGMUYyRjJGNCEiIkYyLChGMEY7KihGOUYySSNSMEdGKEYyRjRGMkYyKiZGNUYyRjFGMkY7RjJGMUY7RjNGOyktSSJTR0YoNiMqJEY0RjtGNUYyNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZGZXhwckdGKCwmKi4sKiomSSJ0R0YoIiIiKUkieEdGKCIiI0YzRjMqKEY2RjNGMkYzRjVGM0YzRjJGM0Y1RjNGMywqKigiIiRGM0YyRjNGNEYzRjNGN0YzKiZGOkYzRjJGM0YzRjUhIiJGMywoKihGNkYzRjJGM0Y0RjNGPCooRjpGM0kjUjBHRihGM0Y1RjNGM0YyRjxGMylGNSIiJUY8KS1JIlNHRig2I0Y1RjZGM0YyRjxGMyooRjpGMylGQEY2RjNGNEY8RjM3I0Yu

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

LCQqKCw0KigiIiQiIiIpSSNSMEc2IkYmRidJInhHRipGJ0YnKioiIz1GJylGKSIiI0YnSSNSMUdGKkYnRitGJ0YnKioiIiVGJ0YuRidJInRHRipGJ0YrRidGJyoqIiNGRidGKUYnKUYwRi9GJ0YrRidGJyosIiInRidGKUYnRjBGJ0YzRidGK0YnRicqKiIiKkYnRjZGJ0YzRidGK0YnRicqKEYvRidGLkYnRjNGJ0YnKipGOEYnRilGJ0YwRidGM0YnISIiKihGOkYnRjZGJ0YzRidGPUYnRjNGPSwoKiQpRitGL0YnRidGK0YnRidGJ0Y9Rj0=

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. We first want to isolate in the numerator powers of x larger than 3 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+SSdudW1wb3NHRigsJioqSSJ0R0YoISIiLCgqJClJI1IwR0YoIiIjIiIiRjcqJkY1RjdGMEY3RjcqJClGMEY2RjdGN0Y3RjVGNylJInhHRigiIiZGN0YxKihGMEYxLDQqKEY0RjdJI1MyR0YoRjdGMEY3RjcqKEY1RjdGQUY3RjpGN0Y3KiZGQUY3KUYwIiIkRjdGNyomIiIlRjcpRjVGRUY3RjcqKCIiJ0Y3RjRGN0kjUjFHRihGN0Y3KihGR0Y3RjRGN0YwRjdGMSoqRkVGN0Y1RjdGS0Y3RjBGN0Y3KihGNkY3RjVGN0Y6RjdGN0YzRjFGNylGPEZHRjdGMTcjRi4=

res:=res+numpos/(1+x+x^2)^2: numneg:=num-numpos: Fexpr:=subs(x=1/x,numneg): Here, we denote dSj = jj D(S)(jj), and dSjinv the counterpart for 1/jj 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)=Sjsol,x=1/jj,Fexpr)/(1-1/jj*x)^2+1/jj*subs(diff(S(x), x)=dSjinv*jj, S(x)=Sjsol,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)=Sjsol,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));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpyZXNub25uZWdHRigsKCouIyIiIiIiJEYxLCYqJiIiI0YxSSNSMEdGKEYxRjFJInRHRihGMUYxLCoqJkY2RjFJInhHRihGMUYxKihGNUYxRjdGMUY6RjFGMUY0RjFGN0YxRjFJI0IxR0YoRjEpRjdGNSEiIiwoKiQpRjpGNUYxRjFGOkYxRjFGMUY+Rj4qLEYzRjEsJkY5RjFGN0Y+RjFJI0IyR0YoRjFGPUY+Rj9GPkYxKigsZHEqLEY1RjEpRjZGNUYxSSNTMkdGKEYxRj1GMSlGOkYyRjFGMSosRjVGMUY2RjFGSUYxKUY3RjJGMUZKRjFGMSoqRjVGMUZJRjEpRjciIiVGMUZKRjFGMSooKUY2RjJGMUY3RjFGSkYxRjEqLCIjN0YxRkhGMUkjUjFHRihGMUY3RjFGSkYxRj4qLEYyRjFGSEYxRklGMUY9RjFGQUYxRjEqKiIjOEYxRkhGMUY9RjFGSkYxRj4qLCIjPUYxRjZGMSlGVEY1RjFGN0YxRkpGMUY+KiwiIipGMUY2RjFGVEYxRj1GMUZKRjFGPiosRjJGMUY2RjFGSUYxRkxGMUZBRjFGMSoqRjJGMUY2RjFGTEYxRkpGMUYxKipGZm5GMUZaRjFGPUYxRkpGMUY+KipGMkYxRklGMUZORjFGQUYxRjEqKkZPRjFGUUYxRjdGMUZBRjFGMSosRjVGMUZIRjFGSUYxRj1GMUY6RjFGMSoqIiM6RjFGSEYxRj1GMUZBRjFGPioqRjJGMUZIRjFGN0YxRkpGMUY+KixGWUYxRjZGMUZaRjFGN0YxRkFGMUY+KixGZm5GMUY2RjFGVEYxRj1GMUZBRjFGMSosRjJGMUY2RjFGVEYxRjdGMUZKRjFGPiosRjVGMUY2RjFGSUYxRkxGMUY6RjFGMSoqIiImRjFGNkYxRkxGMUZBRjFGMSoqRmZuRjFGWkYxRj1GMUZBRjFGPioqRmZuRjFGVEYxRkxGMUZBRjFGMSoqRjVGMUZJRjFGTkYxRjpGMUYxKihGUUYxRjdGMUY6RjFGPiooRjVGMUZRRjFGQUYxRjEqLEZTRjFGSEYxRlRGMUY3RjFGOkYxRj4qKiIiJ0YxRkhGMUZURjFGQUYxRjEqKEZIRjFGSUYxRj1GMUYxKipGV0YxRkhGMUY9RjFGOkYxRj4qKkYyRjFGSEYxRjdGMUZBRjFGPiosRllGMUY2RjFGWkYxRjdGMUY6RjFGPiosRjJGMUY2RjFGVEYxRj1GMUY6RjFGPiooRjZGMUZJRjFGTEYxRjEqKkYyRjFGNkYxRkxGMUY6RjFGMSooRjZGMUY9RjFGQUYxRjEqKkZmbkYxRlpGMUY9RjFGOkYxRj4qKkZdcEYxRlRGMUZMRjFGOkYxRjEqKkYyRjFGVEYxRj1GMUZBRjFGMSomRklGMUZORjFGMSooRjVGMUZRRjFGN0YxRjEqKkZdcEYxRkhGMUZURjFGN0YxRjEqKEY1RjFGSEYxRj1GMUY+KipGZW9GMUZIRjFGN0YxRjpGMUY+KipGXm9GMUY2RjFGVEYxRj1GMUYxKixGZm5GMUY2RjFGVEYxRjdGMUY6RjFGPiooRjJGMUY2RjFGTEYxRjEqKEZmbkYxRlRGMUZMRjFGMSomRkhGMUY3RjFGPiomRjZGMUY9RjFGMSooRjJGMUZURjFGPUYxRjFGMUY9Rj4pRj9GNUY+Rj43I0Yu

We have introduced the series B1 and B2 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);

LCYqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomRiVGJylGKSIiJUYnRis=

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);

LCgqJiIiJiIiIilJInRHNiIiIiVGJSEiIiomIiIjRiUpRiciIiRGJUYqKiYjRiVGLEYlKUYnRixGJUYq

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: mostly positive op(7,eqSdec); factor(series(%,x,2));

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

KylJInhHNiIsKCokKUkjUjBHRiQiIiMiIiIhIiIqJkYoRipJInRHRiRGKkYrKiQpRi1GKUYqRishIiMqJkYtRissMComRilGKilGKCIiJEYqRioqJkYnRipGLUYqRioqKiIiJ0YqRihGKkkjUjFHRiRGKkYtRipGKyooRjhGKkYoRipGL0YqRioqKEY1RipGOUYqRi9GKkYrKiQpRi1GNUYqRipGLEYqRipGKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqIiIh

res:=res+op(7,eqSdec)-convert(%,polynom): We are done 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));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzk=

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

PCtJI0IxRzYiSSNCMkdGJEkjUDBHRiRJI1IwR0YkSSNSMUdGJEkjUzJHRiRJInRHRiRJInhHRiQtSSJTR0YkNiNGKw==

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

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

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

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

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)));

KycsKEkieEc2IiIiIiNGJiIiI0YmKiYsJComRidGJl4jRiZGJkYmRiYpIiIkRidGJiEiIiwkKjAjRiYiI09GJiwmKiZGLEYmRi1GJkYmRi5GJkYmLCoqKEYuRiZJI1MyR0YlRiYpSSJ0R0YlRihGJkYmKihGLkYmSSNSMEdGJUYmRjpGJkYmKiZGLkYmRjlGJkYmKiZGKEYmSSNCMUdGJUYmRiZGJiwmKiZGKEYmRjxGJkYmRjpGJkYmLCgqKEYsRiZGLUYmRjpGJkYmRkFGJkY6RiZGJiwoRkNGJkZBRi9GOkYvRiYpRjpGLkYvRiZGLy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIh

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTMnNvbEdGKCwkKigjIiIiIiIkRjEsKCooRjJGMUkjUjBHRihGMUkidEdGKEYxRjEqJkYyRjEpRjYiIiNGMUYxKiZGOUYxSSNCMUdGKEYxRjFGMUY4ISIiRjw3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZQMHNvbEdGKCwkKiZJInRHRighIiIsMiooIiIlIiIiSSNSMEdGKEY1KUYwIiIjRjVGMSooIiIqRjUpSSNSMUdGKEY4RjVGMEY1RjUqKCIiJEY1RjxGNUY3RjVGMSokKUYwRj5GNUY1KiZJI0IxR0YoRjVGMEY1RjUqKEY4RjVJI0IyR0YoRjVGNkY1RjEqJkZERjVGMEY1RjEqJkY2RjVGMEY1RjFGNUYxNyNGLg==

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

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

We have reached an equation with one catalytic variable for S(x) eqScat:=eqSc: 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));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz0=

We have reached an equation with one catalytic variable for S(x) eqScat:=S(x)^3*(-(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)/x^2) +S(x)^2*(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)*(-2*t*x^2+3*R0*x-t)/(x^2*t*(x^2+x+1)) +S(x)*(-B1+(2*R0+t)*B2/t-9*R1^2-(3*(2*R0*t*x^4+t^2*x^4+2*R0*t*x^3-t^2*x^3+6*R0*t*x^2+t^2*x^2-t*x^3+2*R0*t*x+2*R0*x^2-t^2*x+2*R0*t+t^2-t*x))*R1/(t*x*(x^2+x+1)) -(R0^2*t^2*x^8+R0*t^3*x^8+4*t^4*x^8+4*R0^2*t^2*x^7-10*R0*t^3*x^7+10*t^4*x^7+13*R0^2*t^2*x^6-32*R0*t^3*x^6-R0*t^2*x^7+17*t^4*x^6+2*t^3*x^7+28*R0^2*t^2*x^5-52*R0*t^3*x^5-10*R0*t^2*x^6+20*t^4*x^5+32*R0^2*t^2*x^4+6*R0^2*t*x^5-56*R0*t^3*x^4-7*R0*t^2*x^5+18*t^4*x^4+4*t^3*x^5-t^2*x^6+28*R0^2*t^2*x^3+2*R0^2*t*x^4-38*R0*t^3*x^3-17*R0*t^2*x^4+3*R0*t*x^5+12*t^4*x^3+13*R0^2*t^2*x^2+6*R0^2*t*x^3-2*R0^2*x^4-16*R0*t^3*x^2-7*R0*t^2*x^3+7*t^4*x^2+2*t^3*x^3-t^2*x^4+4*R0^2*t^2*x-4*R0*t^3*x-6*R0*t^2*x^2+R0*t*x^3+2*t^4*x+R0^2*t^2+R0*t^3-R0*t^2*x+t^4)/(t^2*x^2*(x^2+x+1)^2)) +(2*R0^2*x^2+2*R0*t*x^2-t^2*x^2+3*R0*t*x-2*R0^2-2*R0*t-2*t^2)*B1/(3*t^2*(x^2+x+1)) -(2*R0+t)*x*(-t*x+R0)*B2/(t^2*(x^2+x+1)) +(9*(-t*x+ R0)*x*R1^2)/(t*(x^2+x+1)) +(3*(2*R0*t*x^4+t^2*x^4+2*R0*t*x^3-t^2*x^3+6*R0*t*x^2+t^2*x^2-t*x^3+2*R0*t*x+2*R0*x^2-t^2*x+2*R0*t+t^2-t*x))*(-t*x+R0)*R1/(t^2*(x^2+x+1)^2)+(-t*x+R0)*(R0^2*t*x^6+R0*t^2*x^6+t^3*x^6+3*R0^2*t*x^5-2*R0*t^2*x^5+t^3*x^5+3*R0^2*t*x^4-7*R0*t^2*x^4-R0*t*x^5+2*t^3*x^4+6*R0^2*t*x^3-10*R0*t^2*x^3-3*R0*t*x^4+t^3*x^3+3*R0^2*t*x^2+2*R0^2*x^3-7*R0*t^2*x^2-3*R0*t*x^3+2*t^3*x^2+3*R0^2*t*x-2*R0*t^2*x-3*R0*t*x^2+t^3*x+R0^2*t+R0*t^2-R0*t*x+t^3)/(x*t^2*(x^2+x+1)^2);

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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);

LCYqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomRiVGJylGKSIiJUYnRis=

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);

LCgqJiIiJiIiIilJInRHNiIiIiVGJSEiIiomIiIjRiUpRiciIiRGJUYqKiYjRiVGLEYlKUYnRixGJUYq

indets(eqScat)

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSJ0R0YkSSJ4R0YkLUkiU0dGJDYjRik=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTU2hHRigvLUkiU0dGKDYjSSJ4R0YoLCZJI1NoR0YoIiIiKigjRjUiIiRGNSwoKihGOEY1SSNSMEdGKEY1SSJ0R0YoISIiRjUqJiIiI0Y1RjJGNUY9KiRGMkY9Rj1GNSwoRjJGNUZARjVGNUY1Rj1GNTcjRi4=

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

IiIh

This equation is almost symmetric in x and 1/x --- but observe what happens for the constant term 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)))

IiIi

IiIi

ISIi

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

IiRiIw==

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);

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

nops(factor(%))

IiMjKg==

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

IiIk

IiIo

There is a factor y^i in the coefficient of Sb^i ldegree(coeff(eqSbcaty,Sb,3),y); ldegree(coeff(eqSbcaty,Sb,1),y); ldegree(coeff(eqSbcaty,Sb,0),y);

IiIk

IiIi

IiIh

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+SSRjbzNHRigsJCouIiNGIiIiKUkidEdGKCIiI0YxLCZJInlHRihGMUYxRjFGMSwmRjZGMSIiJCEiIkYxLCgqJkYzRjFGNkYxRjFGM0YxRjFGMUYxLCgqKEY4RjFGM0YxRjZGMUYxRjNGOUYxRjlGMUY5NyNGLg==

LCwqKCIjIikiIiIpSSJ5RzYiIiIlRiUpSSJ0R0YoRilGJSEiIioqIiNhRiUpRisiIiRGJSwmKiYiIiNGJUYrRiVGJUYlRixGJSlGJ0YwRiVGJSoqIiNGRiUpRitGM0YlLCgqJiIjOUYlRjdGJUYlKiYiIidGJUYrRiVGJUYlRiVGJSlGJ0YzRiVGJSosRi5GJUY3RiUsJkYrRiVGJUYlRiVGMUYlRidGJUYlKihGJEYlRjdGJSlGP0YzRiVGLA==

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

KigpLCZJInRHNiIiIiJGJ0YnIiIjRicpSSJ5R0YmIiIlISIiKUYlRihGLA==

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

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

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

KigpLCZJInRHNiIiIiJGJ0YnIiIjRicpSSJ5R0YmIiIlISIiKUYlRihGLA==

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

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

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);

KiQpSSJ5RzYiIiIjISIi

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcVN0Y2F0ekdGKCwsKiwiI0YiIiIpSSJ0R0YoIiIjRjEsKComRjRGMUYzRjFGMUkiekdGKEYxRjFGMUYxLCgqJiIjNUYxRjNGMUYxKiYiIiRGMUY3RjEhIiJGMUYxRjEpSSNTdEdGKEY8RjFGMSomLCgqJiwoKiYiJDsjRjFGMkYxRjEqJkYwRjEpRjdGNEYxRj0qJiIjYUYxRjNGMUYxRjEpSSNSMEdGKEY0RjFGMSoqRjBGMUYzRjEsMiooIiInRjFJI1IxR0YoRjFGM0YxRjEqKEZPRjFGUEYxRjdGMUY9KiZGT0YxRjJGMUYxKihGNEYxRjNGMUY3RjFGMSokRkdGMUY9KiZGNEYxSSNCMkdGKEYxRjFGM0YxRjdGMUYxRktGMUYxKigiIipGMUYyRjEsOComRjBGMSlGUEY0RjFGMSooRjBGMUZQRjFGM0YxRj0qKEZYRjFGUEYxRjdGMUYxKiYiIiZGMUYyRjFGMUZTRj0qJkY8RjFJI0IxR0YoRjFGMSomRjxGMUZWRjFGPSomRlhGMUZQRjFGPSomRk9GMUYzRjFGMSomRjRGMUY3RjFGPUYxRjFGMUY9RjFGP0YxRjEqJiwqKiYiI3NGMUYyRjFGMSomRlhGMUZHRjFGPSomIiM9RjFGW29GMUYxKiZGZm9GMUYzRjFGMUYxRkpGMUYxKipGWEYxRjNGMSw0Rk5GMUZRRj1GUkYxRlNGMUZURj0qJkY0RjFGW29GMUYxRlVGMUYzRjFGN0YxRjFGS0YxRjEqJkYyRjEsOiomIiMiKUYxRmVuRjFGPSooRl5wRjFGUEYxRjNGMUYxKihGMEYxRlBGMUY3RjFGPUZobkYxKihGOkYxRjNGMUY3RjFGMSomRjxGMUZHRjFGPSomRlhGMUZbb0YxRjEqJkZYRjFGVkYxRjEqJkYwRjFGUEYxRjFGXm9GPSomIiIlRjFGN0YxRjFGNEY9RjFGMTcjRi4=

IiNg

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSNTdEdGJEkidEdGJEkiekdGJA==

Let us check this compact equation factor(subs(St=1/(x-1/x)*((x+1/x+1)*S(x)-R0/t+2*x/3+1/x/3), z=t*(x+1+1/x)+(t+1)/(x+1+1/x), eqStcatz)/eqScat)

LCQqLCIjRiIiIilJInRHNiIiIiNGJSwoKiQpSSJ4R0YoRilGJUYlRi1GJUYlRiVGJSwmRi1GJUYlISIiRi8sJkYtRiVGJUYlRi9GJQ==

eqStcatz:=27*t^2*(2*t + z + 1)*(10*t - 3*z + 1)*St^3 + ((216*t^2 - 27*z^2 + 54*t)*R0^2 + 27*t*(6*R1*t - 6*R1*z + 6*t^2 + 2*t*z - z^2 + 2*B2 + t + z)*R0 - 9*t^2*(27*R1^2 - 27*R1*t + 9*R1*z + 5*t^2 - 2*t*z + 3*B1 - 3*B2 - 9*R1 + 6*t - 2*z + 1))*St + (72*t^2 - 9*z^2 + 18*B1 + 18*t)*R0^2 + 9*t*(6*R1*t - 6*R1*z + 6*t^2 + 2*t*z - z^2 + 2*B1 + 2*B2 + t + z)*R0 + t^2*(-81*R1^2 + 81*R1*t - 27*R1*z + 5*t^2 + 10*t*z - 3*z^2 + 9*B1 + 9*B2 + 27*R1 - 6*t + 4*z - 2)

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

zx:=t*(x+1+1/x)+(t+1)/(x+1+1/x); StS:=1/(x-1/x)*((x+1/x+1)*S(x)-R0/t+2*x/3+1/x/3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN6eEdGKCwmKiZJInRHRigiIiIsKEkieEdGKEYxKiRGMyEiIkYxRjFGMUYxRjEqJiwmRjBGMUYxRjFGMUYyRjVGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTdFNHRigqJiwmSSJ4R0YoIiIiKiRGMCEiIkYzRjMsKiomLChGMEYxRjJGMUYxRjFGMS1JIlNHRig2I0YwRjFGMSomSSNSMEdGKEYxSSJ0R0YoRjNGMyomIyIiIyIiJEYxRjBGMUYxKiYjRjFGQEYxRjBGM0YxRjE3I0Yu

StS

KiYsJkkieEc2IiIiIiokRiQhIiJGKEYoLCoqJiwoRiRGJkYnRiZGJkYmRiYtSSJTR0YlNiNGJEYmRiYqJkkjUjBHRiVGJkkidEdGJUYoRigqJiMiIiMiIiRGJkYkRiZGJiomI0YmRjVGJkYkRihGJkYm

End of the first session, move to Part 2 for the algebraicity !

JSFH