This Maple session accompanies the paper "Three quadrant walks by invariants" and considers the case of diagonal steps.The proofs are in Section 8.2. restart:

Functional equation. The series Am(1/x) is negative in x K:=1-t*(1/x+x)*(y+1/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYiIiJGLyooSSJ0R0YoRi8sJiokSSJ4R0YoISIiRi9GNEYvRi8sJkkieUdGKEYvKiRGN0Y1Ri9GL0Y1NyNGLg==

eqC:=-K*C(x,y)+1-t/y*(x+1/x)*Cm(1/x)-t/x*(y+1/y)*Cm(1/y)-t/x/y*C00;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsLComLCYiIiJGMSooSSJ0R0YoRjEsJiokSSJ4R0YoISIiRjFGNkYxRjEsJkkieUdGKEYxKiRGOUY3RjFGMUY3RjEtSSJDR0YoNiRGNkY5RjFGN0YxRjEqKkYzRjFGOUY3RjRGMS1JI0NtR0YoNiNGNUYxRjcqKkYzRjFGNkY3RjhGMS1GQDYjRjpGMUY3KipGM0YxRjZGN0Y5RjdJJEMwMEdGKEYxRjc3I0Yu

C00ser:=proc(n) coeff(coeff(Cser(n),x,0),y,0): end: Cmser:=proc(n) local i: local res:res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: Cser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(C(x,y)=Cser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),C00=C00ser(n-1),eqC+C(x,y)),t,n+1)),polynom): fi: end: map(expand,Cser(4));

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Functional equation. The series Am(1/x) is negative in x K:=1-t*(1/x+x)*(y+1/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYiIiJGLyooSSJ0R0YoRi8sJiokSSJ4R0YoISIiRi9GNEYvRi8sJkkieUdGKEYvKiRGN0Y1Ri9GL0Y1NyNGLg==

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

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A00ser:=proc(n) coeff(coeff(Aser(n),x,0),y,0): end: Amser:=proc(n) local i: local res:res:=coeff(convert(Aser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: Aser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(A(x,y)=Aser(n-1),Am(1/x)=Amser(n-1),Am(1/y)=subs(x=y,Amser(n-1)),A00=A00ser(n-1),eqA+A(x,y)),t,n+1)),polynom): fi: end: map(expand,Aser(4));

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We separate in A the contribution of two regions: A(x,y)= U(1/x^2,xy)/x^2 + D(xy) + U(1/y^2,xy)/y^2 U denotes the part of A above the diagonal, DD the diagonal itself.Note that the "corner" of the upper part is now at (-2,0) User:=proc(n) local i, j: collect(add(add(coeff(coeff(convert(Aser(n),polynom),x,i),y,j)*x^((j-i-2)/2)*y^(j),i=-n-2..j-2),j=0..n+4),t,normal): end: User(5);

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Dser:=proc(n) local i: collect(add(coeff(coeff(convert(Aser(n),polynom),x,i),y,i)*y^(i),i=0..n+4),t,normal):end: Dser(3);

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Aexpr:=DD(x*y)+1/x^2*U(1/x^2,x*y)+1/y^2*U(1/y^2,x*y);

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Steps:=(x+1/x)*(y+1/y); tSteps:=expand(subs(y=y/x,x=1/sqrt(x),Steps));

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Equation for D(xy) eqD:=subs(x=1,-DD(x*y)+2/3+t*(x*y+1/x/y)*DD(x*y)-t/x/y*DD(0)+2*t*x/y/x^2*(U(0,x*y)-U(0,0)))

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n:=7:normal(series( subs(U(0,y)=subs(x=0,User(n)),U(0,0)=subs(x=0,y=0,User(n)), DD(y)=Dser(n), DD(0)=subs(y=0,Dser(n)),eqD),t,n))

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eqUinit:=-1/x^2*U(1/x^2,x*y)+1/3/x^2+t*Steps/x^2*U(1/x^2,x*y)+t*y/x*DD(x*y)-t/x/y*(1/x^2*U(1/x^2,0)) -t*x/y* (1/x^2*U(1/x^2,0)+1/x^2*U(0,x*y)-1/x^2*U(0,0))

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eqU:=collect(subs({x=1/sqrt(x),y=sqrt(x)*y},eqUinit),[U,DD],factor)

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expand(coeff(%,U(x,y))/x)

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eqUfinal:=collect(subs(isolate(eqD,U(0,y)),eqU*y*2),[U,DD],factor);

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collect(coeff(%,DD(y))/y,[x,y],factor)

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n:=10:normal(series( subs(U(x,y)=User(n), U(x,0)=subs(y=0,User(n)), DD(y)=Dser(n), DD(0)=subs(y=0,Dser(n)),eqUfinal),t,n))

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The companion model is the x/y-symmetric of Gessel's model tSteps;

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Krg:=1-t*%;

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eqQ:=-Krg*Q(x,y)+1-t/y/x*Q(0,y)-t/x/y*Q(x,0)+t/x/y*Q(0,0)-t/y*Q(x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLiomLCYqJiwqKiZJInhHRigiIiJJInlHRihGNUY1KiRGNiEiIkY1RjZGNSomRjRGOEY2RjhGNUY1SSJ0R0YoRjVGOEY1RjVGNS1JIlFHRig2JEY0RjZGNUY4RjVGNSoqRjpGNUY2RjhGNEY4LUY8NiQiIiFGNkY1RjgqKkY6RjVGNEY4RjZGOC1GPDYkRjRGQUY1RjgqKkY6RjVGNEY4RjZGOC1GPDYkRkFGQUY1RjUqKEY6RjVGNkY4RkNGNUY4NyNGLg==

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(0,0)=subs(x=0,y=0,Qser(n-1)),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)),eqQ+Q(x,y)),t,n+1)),polynom): fi: end: map(expand,Qser(4));

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

Two pairs of invariants (Table 3) I0x:=subs(y=x,y/t/(1+y)^2+t*(1+y)^2/y);J0y:=subs(x=y,x+1/x-t*x^2-t/x^2+2*t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMHhHRigsJiooSSJ4R0YoIiIiSSJ0R0YoISIiKSwmRjBGMUYxRjEiIiNGM0YxKihGMkYxRjRGMUYwRjNGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMHlHRigsLEkieUdGKCIiIiokRi8hIiJGMComSSJ0R0YoRjApRi8iIiNGMEYyKiZGNEYwRjVGMkYyKiZGNkYwRjRGMEYwNyNGLg==

ratio0:=factor((I0x-J0y)/Krg);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzBHRigsJCoqSSJ5R0YoISIiLCwqKEkidEdGKCIiIkkieEdGKEY1KUYwIiIjRjVGNSomRjRGNSlGNkY4RjVGNSomRjRGNUY3RjVGNSomRjRGNUY2RjVGNSomRjZGNUYwRjVGMUY1RjRGMSksJkY2RjVGNUY1RjhGMUYxNyNGLg==

decoupling fx:=x/t/(1+x): gy:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNneUdGKCwkKiRJInlHRighIiJGMTcjRi4=

factor(x*y-fx-gy);factor(%/Krg);

KiosLCooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YpRidGK0YnRicqJkYlRidGKUYnRicqJkYpRidGLEYnISIiRiVGJ0YnRiVGMCwmRilGJ0YnRidGMEYsRjA=

LCQqKEkieEc2IiIiIkkidEdGJSEiIiwmRiRGJkYmRiZGKEYo

I1x:=t*(1+x)*Q(x,0)-fx;J1y:=-t*Q(0,y)+gy+t*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMXhHRigsJiooSSJ0R0YoIiIiLCZJInhHRihGMUYxRjFGMS1JIlFHRig2JEYzIiIhRjFGMSooRjNGMUYwISIiRjJGOUY5NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMXlHRigsKComSSJ0R0YoIiIiLUkiUUdGKDYkIiIhSSJ5R0YoRjEhIiIqJEY2RjdGNyomRjBGMS1GMzYkRjVGNUYxRjE3I0Yu

factor(I1x-J1y); factor(subs(isolate(eqQ,Q(x,0)),%))

KiosNioqLUkiUUc2IjYkSSJ4R0YnIiIhIiIiKUkidEdGJyIiI0YrKUYpRi5GK0kieUdGJ0YrRisqLEYuRitGJUYrRixGK0YpRitGMEYrRisqKkYsRistRiY2JEYqRjBGK0YwRitGKUYrRisqKkYsRistRiY2JEYqRipGK0YwRitGKUYrISIiKihGLEYrRiVGK0YwRitGKyooRixGK0YzRitGMEYrRisqKEYsRitGNkYrRjBGK0Y4KiZGLUYrRilGK0YrKiZGKUYrRjBGK0Y4Ri1GK0YrRi1GOCwmRilGK0YrRitGOEYwRjg=

KiwsLCooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YpRidGK0YnRicqJkYlRidGKUYnRicqJkYpRidGLEYnISIiRiVGJ0YnLCgqKi1JIlFHRiY2JEYpRixGJ0YlRidGKUYnRixGJ0YnKihGM0YnRiVGJ0YsRidGJ0YnRidGJ0YlRjAsJkYpRidGJ0YnRjBGLEYw

ratio1:=factor(%/Krg);collect(%, Q,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzFHRigsJCoqSSJ4R0YoIiIiLCgqKi1JIlFHRig2JEYwSSJ5R0YoRjFJInRHRihGMUYwRjFGN0YxRjEqKEY0RjFGOEYxRjdGMUYxRjFGMUYxRjghIiIsJkYwRjFGMUYxRjpGOjcjRi4=

LCYqKEkieEc2IiIiIkkieUdGJUYmLUkiUUdGJTYkRiRGJ0YmISIiKihGJEYmSSJ0R0YlRissJkYkRiZGJkYmRitGKw==

eqUfinal

LCwqKiIiIyIiIkkidEc2IkYlLCZJInhHRidGJUYlRiVGJS1JIlVHRic2JEYpIiIhRiUhIiIqJiwsKipGJEYlRiZGJSlGKUYkRiUpSSJ5R0YnRiRGJUYlKipGJEYlRiZGJUYpRiVGM0YlRiUqKEYkRiVGJkYlRilGJUYlKihGJEYlRilGJUY0RiVGLiomRiRGJUYmRiVGJUYlLUYrNiRGKUY0RiVGJSomRiZGJS1JI0RER0YnNiNGLUYlRi4qJiwqRjVGJSomRiZGJUYzRiVGJUYmRiVGNEYuRiUtRj02I0Y0RiVGJSooI0YkIiIkRiVGKEYlRjRGJUYl

co:=expand(coeff(%,DD(y))/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwqKioiIiMiIiJJInhHRihGMUkieUdGKEYxSSJ0R0YoRjFGMUYxISIiKiZGM0YxRjRGMUYxKiZGM0Y1RjRGMUYxNyNGLg==

ct:=(2/3)*(x+1)*y

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjdEdGKCwkKigjIiIjIiIkIiIiLCZJInhHRihGM0YzRjNGM0kieUdGKEYzRjM3I0Yu

Ysol:=factor([solve(1-t*tSteps,y)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLCZJInhHRihGMiokKSwkKiZGNUYyLCoqKCIiJUYyKUkidEdGKEYzRjIpRjVGM0YyRjIqKCIiKUYyRj1GMkY1RjJGMiomRjxGMkY9RjJGMkY1ISIiRjJGQ0YxRjJGMkYyRj5GQ0Y1RkMsJkY1RjJGMkYyRkNGMiwkKixGMUYyLCZGNUZDRjZGMkYyRj5GQ0Y1RkNGREZDRkM3I0Yu

Gform:=proc(d,e,f) add(a[i]*y^i,i=0..d)+add(b[i]/(y-beta[i]),i=1..e) +add(c[i]/(y-gamma[i])^2,i=1..f): end: Gform(2,3,4)

LDYqJiZJImFHNiI2IyIiIyIiIilJInlHRiZGKEYpRikqJiZGJTYjRilGKUYrRilGKSZGJTYjIiIhRikqJiZJImJHRiZGLkYpLCZGK0YpJkklYmV0YUdGJkYuISIiRjhGKSomJkY0RidGKSwmRitGKSZGN0YnRjhGOEYpKiYmRjQ2IyIiJEYpLCZGK0YpJkY3Rj9GOEY4RikqJiZJImNHRiZGLkYpKSwmRitGKSZJJmdhbW1hRyUqcHJvdGVjdGVkR0YuRjhGKEY4RikqJiZGRUYnRikpLCZGK0YpJkZJRidGOEYoRjhGKSomJkZFRj9GKSksJkYrRikmRklGP0Y4RihGOEYpKiYmRkU2IyIiJUYpKSwmRitGKSZGSUZXRjhGKEY4Rik=

d:=0: e:=1: f:=1:collect(numer(factor(expand(rationalize(subs(y=Ysol[1],co*Gform(d,e,f)-ct)-subs(y=Ysol[2],co*Gform(d,e,f)-ct))))/sqrt(-x*(4*t^2*x^2+8*t^2*x+4*t^2-x))),x,factor): coeffs(%,x): sol:=solve({%}, [seq(a[i], i=0..d), seq(b[i], i=1..e), seq(beta[i], i=1..e), seq(c[i], i=1..f), seq(gamma[i], i=1..f)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig3IzcnLyZJImFHRig2IyIiISwkKiYjIiIjIiIkIiIiSSJ0R0YoISIiRjovJkkiYkdGKDYjRjpGNC8mSSViZXRhR0YoRkBGQi8mSSJjR0YoRkBGNC8mSSZnYW1tYUdGJkZARkg3I0Yu

Gform(d,e,f)

LCgmSSJhRzYiNiMiIiEiIiIqJiZJImJHRiU2I0YoRigsJkkieUdGJUYoJkklYmV0YUdGJUYsISIiRjFGKComJkkiY0dGJUYsRigpLCZGLkYoJkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRixGMSIiI0YxRig=

Gy:=subs(op(1,sol),Gform(d,e,f));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiYjIiIjIiIkIiIiSSJ0R0YoISIiRjM3I0Yu

The function F(x) Fx:=-factor(expand(rationalize(subs(y=Ysol[1], op(sol),-ct+co*Gform(d,e,f)))));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKigjIiIjIiIkIiIiSSJ0R0YoISIiLCZJInhHRihGM0YzRjNGNUYzNyNGLg==

factor(ct-Gy*co-Fx);

LCQqLCMiIiMiIiQiIiIsLCooSSJ0RzYiRicpSSJ4R0YrRiVGJylJInlHRitGJUYnRicqKEYqRidGLUYnRi5GJ0YnKiZGKkYnRi1GJ0YnKiZGLUYnRi9GJyEiIkYqRidGJ0YqRjMsJkYtRidGJ0YnRjNGL0YzRjM=

Hxy:=factor(%/(1-t*tSteps))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeHlHRigsJCoqIyIiIyIiJCIiIkkieEdGKEYzSSJ0R0YoISIiLCZGNEYzRjNGM0Y2RjM3I0Yu

Fx := 2/(3*t*(x + 1)); Gy := 2/(3*t); Hxy := (2*x)/(3*t*(x + 1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKigjIiIjIiIkIiIiSSJ0R0YoISIiLCZJInhHRihGM0YzRjNGNUYzNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiYjIiIjIiIkIiIiSSJ0R0YoISIiRjM3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeHlHRigsJCoqIyIiIyIiJCIiIkkieEdGKEYzSSJ0R0YoISIiLCZGNEYzRjNGM0Y2RjM3I0Yu

factor(ct-co*Gy-Fx)

LCQqLCMiIiMiIiQiIiIsLCooSSJ0RzYiRicpSSJ4R0YrRiVGJylJInlHRitGJUYnRicqKEYqRidGLUYnRi5GJ0YnKiZGKkYnRi1GJ0YnKiZGLUYnRi9GJyEiIkYqRidGJ0YqRjMsJkYtRidGJ0YnRjNGL0YzRjM=

tSteps

LCoqJkkieEc2IiIiIkkieUdGJUYmRiYqJEYnISIiRiZGJ0YmKiZGJEYpRidGKUYm

factor(%%/(1-t*tSteps))

LCQqKiMiIiMiIiQiIiJJInhHNiJGJ0kidEdGKSEiIiwmRihGJ0YnRidGK0Yn

eqUfinal

LCwqKiIiIyIiIkkidEc2IkYlLCZJInhHRidGJUYlRiVGJS1JIlVHRic2JEYpIiIhRiUhIiIqJiwsKipGJEYlRiZGJSlGKUYkRiUpSSJ5R0YnRiRGJUYlKipGJEYlRiZGJUYpRiVGM0YlRiUqKEYkRiVGJkYlRilGJUYlKihGJEYlRilGJUY0RiVGLiomRiRGJUYmRiVGJUYlLUYrNiRGKUY0RiVGJSomRiZGJS1JI0RER0YnNiNGLUYlRi4qJiwqRjVGJSomRiZGJUYzRiVGJUYmRiVGNEYuRiUtRj02I0Y0RiVGJSooI0YkIiIkRiVGKEYlRjRGJUYl

Sy:=y*DD(y)+Gy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCwmKiZJInlHRigiIiItSSNEREdGKDYjRjBGMUYxKiYjIiIjIiIkRjFJInRHRighIiJGMTcjRi4=

Rx:=2*t*(x+1)*U(x,0)+t*DD(0)-Fx;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNSeEdGKCwoKioiIiMiIiJJInRHRihGMSwmSSJ4R0YoRjFGMUYxRjEtSSJVR0YoNiRGNCIiIUYxRjEqJkYyRjEtSSNEREdGKDYjRjhGMUYxKigjRjAiIiRGMUYyISIiRjNGQEZANyNGLg==

Delta:=collect(discrim(numer(1-t*tSteps),x)/y^2,t,factor);factor(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwoIiIiRi8qKiksJkkieUdGKEYvRi8hIiIiIiNGLyksJkYvRi9GM0YvRjVGLylGM0Y1RjQpSSJ0R0YoRjVGL0YvKipGNUYvRjNGNCwmKiRGOEYvRi9GL0YvRi9GOkYvRjQ3I0Yu

KigsKiomSSJ0RzYiIiIiKUkieUdGJiIiI0YnRicqKEYqRidGJUYnRilGJ0YnRiVGJ0YpISIiRicsKkYkRidGK0YsRiVGJ0YpRixGJ0YoRiw=

Deltapap:=(1-t*(y+1)^2/y)*(1-t*(y-1)^2/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlEZWx0YXBhcEdGKComLCYiIiJGMCooSSJ0R0YoRjApLCZGMEYwSSJ5R0YoRjAiIiNGMEY1ISIiRjdGMCwmRjBGMCooRjJGMCksJkY1RjBGMEY3RjZGMEY1RjdGN0YwNyNGLg==

factor(Deltapap/Delta)

IiIi

factor(Rx^2-Delta*Sy^2);

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

Divisibility by the kernel res:=factor(subs(isolate(eqUfinal,U(x,0)),%)/(1-t*tSteps));

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

Sy;

LCYqJkkieUc2IiIiIi1JI0RER0YlNiNGJEYmRiYqJiMiIiMiIiRGJkkidEdGJSEiIkYm

Rx;Delta;

LCgqKiIiIyIiIkkidEc2IkYlLCZJInhHRidGJUYlRiVGJS1JIlVHRic2JEYpIiIhRiVGJSomRiZGJS1JI0RER0YnNiNGLUYlRiUqKCNGJCIiJEYlRiYhIiJGKEY1RjU=

LCgiIiJGIyoqKSwmSSJ5RzYiRiNGIyEiIiIiI0YjKSwmRiNGI0YnRiNGKkYjKUYnRipGKSlJInRHRihGKkYjRiMqKkYqRiNGJ0YpLCYqJEYtRiNGI0YjRiNGI0YvRiNGKQ==

Ix:=Rx^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCokKSwoKioiIiMiIiJJInRHRihGMywmSSJ4R0YoRjNGM0YzRjMtSSJVR0YoNiRGNiIiIUYzRjMqJkY0RjMtSSNEREdGKDYjRjpGM0YzKigjRjIiIiRGM0Y0ISIiRjVGQkZCRjJGMzcjRi4=

Jy:=Delta*Sy^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKComLCgiIiJGMCoqKSwmSSJ5R0YoRjBGMCEiIiIiI0YwKSwmRjBGMEY0RjBGNkYwKUY0RjZGNSlJInRHRihGNkYwRjAqKkY2RjBGNEY1LCYqJEY5RjBGMEYwRjBGMEY7RjBGNUYwKSwmKiZGNEYwLUkjRERHRig2I0Y0RjBGMComI0Y2IiIkRjBGO0Y1RjBGNkYwNyNGLg==

Watch out: we have to swap x and y in the Gessel invariants I1x; J1y;

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKEYoRiZGJCEiIkYnRi5GLg==

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

Poles at -1 for the x-invariants, and 0 for the y-invariants Ix; I1x;

KiQpLCgqKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJyomRihGJy1JI0RER0YpNiNGL0YnRicqKCNGJiIiJEYnRighIiJGKkY3RjdGJkYn

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKEYoRiZGJCEiIkYnRi5GLg==

Jy; J1y;

KiYsKCIiIkYkKiopLCZJInlHNiJGJEYkISIiIiIjRiQpLCZGJEYkRihGJEYrRiQpRihGK0YqKUkidEdGKUYrRiRGJCoqRitGJEYoRiosJiokRi5GJEYkRiRGJEYkRjBGJEYqRiQpLCYqJkYoRiQtSSNEREdGKTYjRihGJEYkKiYjRisiIiRGJEYwRipGJEYrRiQ=

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

We ignore the rational pair and try to combine I,J and I1, J1 series(Jy,y,2);series(J1y,y,1)

KylJInlHNiIjIiIlIiIqISIjLCYqKCNGJiIiJCIiIkkidEdGJEYtLUkjRERHRiQ2IyIiIUYtRi0qJiMiIilGJ0YtRi4hIiJGNkY2LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRi1GMg==

KydJInlHNiIhIiJGJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIkYs

factor(series(Jy-(4/9*I1y^2),y,2));

KylJInlHNiIjIiIlIiIqISIjLCQqKEYlIiIiLCYqKCIiJEYrKUkidEdGJCIiI0YrLUkjRERHRiQ2IyIiIUYrRitGMSEiIkYrRjBGNkYrRjYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGK0Y1

A1:=-coeff(%,y,-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMUdGKCwkKigjIiIlIiIqIiIiLCYqKCIiJEYzKUkidEdGKCIiI0YzLUkjRERHRig2IyIiIUYzRjNGOSEiIkYzRjhGPkY+NyNGLg==

An invariant with no pole at 0 factor(series(Jy-(4/9*J1y^2+A1*J1y),y,3));

KydJInlHNiIsJCooIyIiIiIiKkYoLC4qKEYpRigpLUkjRERHRiQ2IyIiISIiI0YoKUkidEdGJCIiJUYoRigqKCIjN0YoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YuRi9GKClGMyIiJEYoRigqKCIiKUYoRj5GKC0tJkY5NiNGMTYjSSJRR0YkNiRGMEYwRighIiIqKCIjQ0YoKUYzRjFGKEYtRihGSSomRkFGKEZMRihGSUY0RihGKEZMRklGKEYwLUkiT0dGOjYjRihGKA==

What happened to the x-invariant meanwhile? We see that it has no more pole at -1 factor(series(Ix-(4/9*I1x^2+A1*I1x),x=-1,2))

KycsJkkieEc2IiIiIkYmRiYsJCooI0YmIiIqRiYsLCooRipGJiktSSNEREdGJTYjIiIhIiIjRiYpSSJ0R0YlIiIlRiYhIiIqKCIjQ0YmLUkiVUdGJTYkRjZGMUYmKUY0RjJGJkYmKigiIilGJi1JIlFHRiVGO0YmRjxGJkYmKigiIzdGJkY8RiZGLkYmRiZGNUY2RiZGPEY2RjZGMS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmRiY=

Let us check this using the known values of Q(x,0). This gives the first terms of the unknown series A, n:=16: A0expser:=normal(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)),U(0,0)=subs(x=0,y=0,User(n)),DD(0)=subs(y=0,Dser(n)) ,Ix-(4/9*I1x^2+A1*I1x) ),t,n));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlBMGV4cHNlckdGKCs1SSJ0R0YoIyIiJSIiKiEiIyMhIikiIiQiIiEjISM/RjYiIiMjISRDJUYyRjEjISVfUkYyIiInIyEmX0AlRjIiIikhJlNUJiIjNSEnNy5tIiM3LUkiT0dGJTYjIiIiIiM5NyNGLg==

normal(series(subs(DD(y)=Dser(n), DD(0)=subs(y=0,Dser(n)),Q(0,y)=subs(x=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),U(0,0)=subs(x=0,y=0,User(n)), Jy-(4/9*J1y^2+A1*J1y)),t,n));

KzdJInRHNiIjIiIlIiIqISIjIyEiKSIiJCIiISMhIz9GKyIiIyMhJEMlRidGJiMhJV9SRiciIicjISZfQCVGJyIiKSEmU1QmIiM1ISc3Lm0iIzcjISlXV0p2RiciIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

This is encouraging! Let us check that the ratio is a multiple of xy The kernel disappears from the denominator ratio:=factor(subs(isolate(eqUfinal,U(x,0)), isolate(eqQ,Q(x,0)), Ix-(4/9*I1x^2+A1*I1x) - (Jy-(4/9*J1y^2+A1*J1y)))/(1-t*tSteps)): denom(%)

LCQqKCIiKiIiIkkidEc2IkYlLCZJInhHRidGJUYlRiVGJUYl

This is a multiple of x factor(series(ratio,x,2))

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

and a multiple of y series(ratio,y,1)

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

Ixnul:=Ix-4/9*I1x^2-A1*I1x-A0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZJeG51bEdGKCwqKiQpLCgqKiIiIyIiIkkidEdGKEY0LCZJInhHRihGNEY0RjRGNC1JIlVHRig2JEY3IiIhRjRGNComRjVGNC1JI0RER0YoNiNGO0Y0RjQqKCNGMyIiJEY0RjUhIiJGNkZDRkNGM0Y0RjQqJiMiIiUiIipGNCksJiooRjVGNEY2RjQtSSJRR0YoRjpGNEY0KihGN0Y0RjVGQ0Y2RkNGQ0YzRjRGQyoqRkVGNCwmKihGQkY0KUY1RjNGNEY9RjRGNEYzRkNGNEY1RkNGSUY0RjRJI0EwR0YoRkM3I0Yu

Jynul:=Jy-4/9*J1y^2-A1*J1y-A0

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZKeW51bEdGKCwqKiYsKCIiIkYxKiopLCZJInlHRihGMUYxISIiIiIjRjEpLCZGMUYxRjVGMUY3RjEpRjVGN0Y2KUkidEdGKEY3RjFGMSoqRjdGMUY1RjYsJiokRjpGMUYxRjFGMUYxRjxGMUY2RjEpLCYqJkY1RjEtSSNEREdGKDYjRjVGMUYxKiYjRjciIiRGMUY8RjZGMUY3RjFGMSomIyIiJSIiKkYxKSwoKiZGPEYxLUkiUUdGKDYkIiIhRjVGMUY2KiRGNUY2RjYqJkY8RjEtRlE2JEZTRlNGMUYxRjdGMUY2KipGSkYxLCYqKEZIRjFGO0YxLUZENiNGU0YxRjFGN0Y2RjFGPEY2Rk5GMUYxSSNBMEdGKEY2NyNGLg==

n:=6: normal(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)),U(0,0)=subs(x=0,y=0,User(n)),DD(0)=subs(y=0,Dser(n)) ,Ixnul ),t,n));

KytJInRHNiIjIiIlIiIqISIjLCZJI0EwR0YkISIiIyIiKSIiJEYrIiIhIyEjP0YuIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiRiY=

normal(series(subs(DD(y)=Dser(n), DD(0)=subs(y=0,Dser(n)),Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),U(0,0)=subs(x=0,y=0,User(n)),Jynul),t,n));

Ky1JInRHNiIjIiIlIiIqISIjLCZJI0EwR0YkISIiIyIiKSIiJEYrIiIhIyEjP0YuIiIjIyEkQyVGJ0YmLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiIm

Two roots are known from the roots of Delta. Ixform:=Ix-4/9*(I1-a)*(I1-b); Jyform:=Jy-4/9*(J1-a)*(J1-b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJeGZvcm1HRigsJiokKSwoKioiIiMiIiJJInRHRihGNCwmSSJ4R0YoRjRGNEY0RjQtSSJVR0YoNiRGNyIiIUY0RjQqJkY1RjQtSSNEREdGKDYjRjtGNEY0KigjRjMiIiRGNEY1ISIiRjZGQ0ZDRjNGNEY0KigjIiIlIiIqRjQsJkkjSTFHRihGNEkiYUdGKEZDRjQsJkZJRjRJImJHRihGQ0Y0RkM3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeWZvcm1HRigsJiomLCgiIiJGMSoqKSwmSSJ5R0YoRjFGMSEiIiIiI0YxKSwmRjFGMUY1RjFGN0YxKUY1RjdGNilJInRHRihGN0YxRjEqKkY3RjFGNUY2LCYqJEY6RjFGMUYxRjFGMUY8RjFGNkYxKSwmKiZGNUYxLUkjRERHRig2I0Y1RjFGMSomI0Y3IiIkRjFGPEY2RjFGN0YxRjEqKCMiIiUiIipGMSwmSSNKMUdGKEYxSSJhR0YoRjZGMSwmRk5GMUkiYkdGKEY2RjFGNjcjRi4=

with(gfun): factor(numer(Delta))

KiYsKiomSSJ0RzYiIiIiKUkieUdGJiIiI0YnRicqKEYqRidGJUYnRilGJ0YnRiVGJ0YpISIiRicsKkYkRidGK0YsRiVGJ0YpRixGJw==

The roots of Delta are Catalanish series Ysolens:=map(allvalues,algeqtoseries(numer(Delta),t,y,27,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShZc29sZW5zR0YoNyQrQUkidEdGKCIiIkYxIiIjRjIiIiYiIiQiIzkiIiUiI1VGMyIkSyIiIiciJEglIiIoIiVJOSIiKSIlaVsiIioiJid6OyIjNSImJ3llIiM2Iic3ITMjIiM3IicrSHUiIzgiKFNXbiNGNS1JIk9HRiU2I0YxIiM6K0FGMEYxRjEhIiNGMkYzRjQhIzlGNkY3RjMhJEsiRjlGOkY7ISVJOUY9Rj5GPyEmJ3o7RkFGQkZDISc3ITMjRkVGRkZHIShTV24jRjVGSUZMNyNGLg==

Y0ser:=op(1,%):Y1ser:=op(2,%%): J1y

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

n:=20:aser:=series(subs(Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)),y=Y0ser, J1y),t,16)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc2VyR0YoK0FJInRHRighIiJGMCIiIyIiISIiIkYzRjFGMSIiJSIiJCIjN0Y0IiNKIiImIiQvIiIiJyIkLCQiIigiJWs1IiIpIiUnRyQiIioiJnE+IiIjNSImaSZRIiM2Iic/SzlGNi1JIk9HRiU2I0YzIiM4NyNGLg==

bser:=series(subs(Q(0,y)=subs(x=0,Qser(n)),y=Y1ser,Q(0,0)=subs(x=0,y=0,Qser(n)), J1y),t,16)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic2VyR0YoK0FJInRHRighIiJGMCEiIyIiISIiIkYzRjEiIiMiIiUiIiQhIzdGNSIjSiIiJiEkLyIiIiciJCwkIiIoISVrNSIiKSIlJ0ckIiIqISZxPiIiIzUiJmkmUSIjNiEnP0s5IiM3LUkiT0dGJTYjRjMiIzg3I0Yu

let us check n:=20: factor(series(subs(U(x,0)=subs(y=0,User(n+4)),I1=I1x,Q(0,0)=subs(x=0,y=0,Qser(n+4)), Q(x,0)=subs(y=0,Qser(n+4)), DD(0)=subs(y=0,Dser(n+4)), a=aser, b=bser,Ixform),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

n:=16:factor(series(subs(DD(y)=Dser(n),J1=J1y, Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)), a=aser, b=bser,Jyform),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

We still have to express D0 factor(series(subs(J1=J1y,Jyform),y,3))

KylJInlHNiIsJCooIyIiJSIiKiIiIiwqKigiIiRGKilJInRHRiQiIiNGKi1JI0RER0YkNiMiIiFGKkYqKiZJImFHRiRGKkYvRiohIiIqJkkiYkdGJEYqRi9GKkY3RjBGN0YqRi9GN0YqRjcsJCooI0YqRilGKiwwKihGKUYqKUYxRjBGKilGL0YoRipGKiooIiM3RiotLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRjJGM0YqKUYvRi1GKkYqKigiIilGKkZKRiotLSZGRTYjRjA2I0kiUUdGJDYkRjRGNEYqRjcqKkYoRipGNkYqRjlGKkYuRipGNyooIiNDRipGLkYqRjFGKkY3KiZGTEYqRi5GKkY3RihGKkYqRi5GN0YqRjQtSSJPR0ZGNiNGKkYq

collect(isolate(op(1,%),DD(0)),[Q,a,b],factor)

Ly1JI0RERzYiNiMiIiEsKCooIyIiIiIiJEYrSSJ0R0YlISIiSSJhR0YlRitGKyooRipGK0YtRi5JImJHRiVGK0YrKiYjIiIjRixGKylGLUY0Ri5GKw==

factor(%)

Ly1JI0RERzYiNiMiIiEsJCooIyIiIiIiJEYrLCgqJkkiYUdGJUYrSSJ0R0YlRitGKyomSSJiR0YlRitGMEYrRisiIiNGK0YrKUYwRjMhIiJGKw==

We use known results for Gessel's walks in the quadrant, taking care to exchange x and y

t2T:=(T-1)*(T+3)^3/256/T^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0MlRHRigsJCoqIyIiIiIkYyNGMSwmSSJUR0YoRjFGMSEiIkYxKSwmRjRGMSIiJEYxRjhGMSlGNEY4RjVGMTcjRi4=

with(gfun): Tser:= proc(n) 1+t*op(1,algeqtoseries(subs(T=1+t*P,numer(t^2-t2T)),t,P,n,true)): end: Tser(5); Tser(6)

LCYiIiJGIyomSSJ0RzYiRiMrK0YlIiIlRiMiI08iIiQiJCdSIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiMiIidGI0Yj

LCYiIiJGIyomSSJ0RzYiRiMrK0YlIiIlRiMiI08iIiQiJCdSIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiMiIihGI0Yj

algTs:=numer(s-t2T)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdUc0dGKCwsKiQpSSJUR0YoIiIlIiIiISIiKigiJGMjRjMpRjEiIiRGM0kic0dGKEYzRjMqJiIiKUYzRjdGM0Y0KiYiIz1GMylGMSIiI0YzRjQiI0ZGMzcjRi4=

Introduce Z=sqrt(T) t2Z:=factor(subs(T=Z^2,t2T));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0MlpHRigsJCosIyIiIiIkYyNGMSwmSSJaR0YoRjFGMSEiIkYxLCZGNEYxRjFGMUYxKSwmKiQpRjQiIiNGMUYxIiIkRjFGPEYxKUY0IiInRjVGMTcjRi4=

Zser:=proc(n) series(sqrt(Tser(n)),t,n+1): end : Zser(5); Zser(6)

KytJInRHNiIiIiIiIiEiIiNGJyIjOyIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

Ky1JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKQ==

y in terms of T and an additional series U yTU:=factor(solve(16*T^2*(U^2-T)-x*(-2*T+U+U*T)*(-9*T+U^2+8*T*U+T^2-T*U^2),x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR5VFVHRigsJCosIiM7IiIiKUkiVEdGKCIiI0YxLCYqJClJIlVHRihGNEYxISIiRjNGMUYxLCgqJkYzRjFGOEYxRjEqJkY0RjFGM0YxRjlGOEYxRjksLComRjNGMUY3RjFGOSokRjJGMUYxKigiIilGMUYzRjFGOEYxRjFGNkYxKiYiIipGMUYzRjFGOUY5Rjk3I0Yu

yZU:=factor(subs(T=Z^2,yTU));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR5WlVHRigsJCowIiM7IiIiKUkiWkdGKCIiJUYxLCZJIlVHRihGMUYzISIiRjEsJkY2RjFGM0YxRjEsKComRjZGMSlGMyIiI0YxRjEqJkY8RjFGO0YxRjdGNkYxRjcsKiomRjZGMUYzRjFGMSokRjtGMUYxRjZGMSomIiIkRjFGM0YxRjdGNywqRj9GMUZARjdGNkY3RkFGN0Y3Rjc3I0Yu

The expression of Q(0,yt) QytZU:= 16*Z^2*(U*Z^2-2*Z^2+U)*(U^3*Z^4+U^2*Z^5-U*Z^6-Z^7-16*U^2*Z^4-16*U*Z^5-2*U^3*Z^2-2*U^2*Z^3+82*U*Z^4+18*Z^5+16*U^2*Z^2+16*U*Z^3-128*Z^4+U^3+U^2*Z-17*U*Z^2-81*Z^3)/((Z-1)*(Z+1)*(Z^2+3)^3*(U*Z+Z^2+U-3*Z)*(U+Z)*(U*Z-Z^2-U-3*Z))

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

The correct value of U is the one with CT equal to 1 UserTm1:=proc(n) 1+Tm1*op(1,map(factor,map(allvalues,algeqtoseries(subs(T=1+Tm1,U=1+Tm1*P,numer(y-yTU)),Tm1,P,n+1,true)))): end: UserTm1(4); UserTm1(3)

LCYiIiJGIyomSSRUbTFHNiJGIysvRiUjRiMiIiMiIiEjISIiIiIpRiMsJiomI0YjIiNLRiNJInlHRiZGI0YjI0YjIiM7RiNGKSwoKiYjIiIoIiRHIkYjRjJGI0YsIyIiJkY5RiwqJiNGI0Y5RiMpRjJGKUYjRiMiIiQsKiomIyIjUCIkNyZGI0YyRiNGIyNGOCIkYyNGIyomIyIjNkZERiNGPkYjRiwqJiNGI0ZERiMpRjJGP0YjRiMiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGI0Y7RiNGIw==

LCYiIiJGIyomSSRUbTFHNiJGIystRiUjRiMiIiMiIiEjISIiIiIpRiMsJiomI0YjIiNLRiNJInlHRiZGI0YjI0YjIiM7RiNGKSwoKiYjIiIoIiRHIkYjRjJGI0YsIyIiJkY5RiwqJiNGI0Y5RiMpRjJGKUYjRiMiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGIyIiJUYjRiM=

User:=proc(n) series(subs(Tm1=Tser(n)-1,UserTm1(n)),t,n+1): end: User(6);User(8)

Ky1JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSwmIiRtIkYlKiZGJ0YlSSJ5R0YkRiVGJSIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIp

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSwmIiRtIkYlKiZGJ0YlSSJ5R0YkRiVGJSIiJywoKiZGJ0YlKUYtRidGJUYlKiYiI1NGJUYtRiVGJSIlTT5GJSIiKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM1

We now consider Q(x,0) xTV:=(1-T+3*V+V*T)/V^2/(3+V+T-V*T);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR4VFZHRigqKCwqKiZJIlRHRigiIiJJIlZHRihGMkYyRjEhIiIqJiIiJEYyRjNGMkYyRjJGMkYyKUYzIiIjRjQsKkYwRjRGMUYyRjNGMkY2RjJGNDcjRi4=

xZV:=factor(subs(T=Z^2,xTV));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR4WlZHRigsJCooLCoqJkkiVkdGKCIiIilJIlpHRigiIiNGM0YzKiRGNEYzISIiKiYiIiRGM0YyRjNGM0YzRjNGMylGMkY2RjgsKkYxRjNGN0Y4RjJGOEY6RjhGOEY4NyNGLg==

QxZV := -16*V*Z^3*(V*Z^2-Z^2-V-3)*(V^3*Z^4-2*V^3*Z^3+V^2*Z^4+4*V^3*Z^2+2*V^2*Z^3+V*Z^4-6*V^3*Z-8*V^2*Z^2-2*V*Z^3+Z^4+3*V^3-2*V^2*Z-8*V*Z^2+2*Z^3+7*V^2+2*V*Z+4*Z^2+7*V+6*Z+3)/((V*Z-V-Z-1)^2*(V+1)^2*(Z^2+3)^3*(Z+1)*(Z-1))

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

the correct V is V = O(t) algeqtoseries(subs(T=1+Tm1,V=Tm1*W,numer(xTV-x)),Tm1,W,3,true)

NyMrK0kkVG0xRzYiIyIiIiIiJSIiISwmKiYjIiIiIiM7IiIiSSJ4RzYiIiIiIiIiIyIiIiIjOyEiIiIiIiwoKiYjIiIiIiNLIiIiKUkieEc2IiIiIyIiIiIiIiomIyIiIiIjSyIiIkkieEc2IiIiIiEiIiMiIiIiI2siIiIiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiIiIiQ=

VserTm1:=proc(n) Tm1*op(algeqtoseries(subs(T=1+Tm1,V=Tm1*W,numer(xTV-x)),Tm1,W,n,true)): end: VserTm1(4)

KiZJJFRtMUc2IiIiIistRiMjRiUiIiUiIiEsJiomI0YlIiM7RiVJInhHRiRGJUYlRiwhIiJGJSwoKiYjRiUiI0tGJSlGLiIiI0YlRiUqJkYyRiVGLkYlRi8jRiUiI2tGJUY1LCoqJiNGJSIkRyJGJUYuRiVGJSomIyIiJEY8RiVGNEYlRi8qJiMiIiYiJGMjRiUpRi5GP0YlRiUjRiVGQ0YvRj8tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYoRiU=

Vser:=proc(n) series(subs(Tm1=Tser(n)-1,VserTm1(n)),t,n+1): end: Vser(6);Vser(8)

KytJInRHNiIiIiIiIiMsJkkieEdGJEYlIiIpRiUiIiUsKComRiZGJSlGKEYmRiVGJSomIiM7RiVGKEYlRiUiIyMpRiUiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYp

Ky1JInRHNiIiIiIiIiMsJkkieEdGJEYlIiIpRiUiIiUsKComRiZGJSlGKEYmRiVGJSomIiM7RiVGKEYlRiUiIyMpRiUiIicsKiomIiImRiUpRigiIiRGJUYlKiYiI1tGJUYtRiVGJSomIiRGI0YlRihGJUYlIiRXKkYlRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

Let us check User(10); Zser(10)

KzFJInRHNiIiIiIiIiEiIiNGJyIjOyIiJSwmIiRtIkYlKiZGJ0YlSSJ5R0YkRiVGJSIiJywoKiZGJ0YlKUYtRidGJUYlKiYiI1NGJUYtRiVGJSIlTT5GJSIiKSwqKiZGJ0YlKUYtIiIkRiVGJSomIiNdRiVGMUYlRiUqJiIkXSdGJUYtRiVGJSImd1MjRiUiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNw==

KzFJInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpIiZ3UyMiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNw==

n:=12: normal(series(subs(U=User(n), Z=Zser(n), QytZU),t,1+n))

Ky9JInRHNiIiIiIiIiEsJkkieUdGJEYlIiIjRiVGKSwoKiQpRihGKUYlRiUqJiIiJkYlRihGJUYlIiM2RiUiIiUsKiokKUYoIiIkRiVGJSomIiIqRiVGLEYlRiUqJiIjUEYlRihGJUYlIiMmKUYlIiInLCwqJClGKEYwRiVGJSomIiM5RiVGM0YlRiUqJiIjKClGJUYsRiVGJSomIiRLJEYlRihGJUYlIiQjeUYlIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIzU=

series(subs(x=0,y=t*y,Qser(10)),t,10)

Ky9JInRHNiIiIiIiIiEsJkkieUdGJEYlIiIjRiVGKSwoKiQpRihGKUYlRiUqJiIiJkYlRihGJUYlIiM2RiUiIiUsKiokKUYoIiIkRiVGJSomIiIqRiVGLEYlRiUqJiIjUEYlRihGJUYlIiMmKUYlIiInLCwqJClGKEYwRiVGJSomIiM5RiVGM0YlRiUqJiIjKClGJUYsRiVGJSomIiRLJEYlRihGJUYlIiQjeUYlIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIzU=

Now for Q(x,0) n:=10:normal(series(subs(V=Vser(n),Z=Zser(n),QxZV),t,n+1))

Ky9JInRHNiIiIiIiIiEsJkkieEdGJEYlIiIjRiVGKSwoKiZGKUYlKUYoRilGJUYlKiYiIilGJUYoRiVGJSIjNkYlIiIlLCoqJiIiJkYlKUYoIiIkRiVGJSomIiNJRiVGLEYlRiUqJiIjdEYlRihGJUYlIiMmKUYlIiInLCwqJiIjOUYlKUYoRjBGJUYlKiYiJDciRiVGNEYlRiUqJiIkKFFGJUYsRiVGJSomIiRSKEYlRihGJUYlIiQjeUYlRi4tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

series(subs(y=0,Qser(n)),t,10)

Ky9JInRHNiIiIiIiIiEsJkkieEdGJEYlIiIjRiVGKSwoKiZGKUYlKUYoRilGJUYlKiYiIilGJUYoRiVGJSIjNkYlIiIlLCoqJiIiJkYlKUYoIiIkRiVGJSomIiNJRiVGLEYlRiUqJiIjdEYlRihGJUYlIiMmKUYlIiInLCwqJiIjOUYlKUYoRjBGJUYlKiYiJDciRiVGNEYlRiUqJiIkKFFGJUYsRiVGJSomIiRSKEYlRihGJUYlIiQjeUYlRi4tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

Delta; J1y

LCgiIiJGIyoqKSwmSSJ5RzYiRiNGIyEiIiIiI0YjKSwmRiNGI0YnRiNGKkYjKUYnRipGKSlJInRHRihGKkYjRiMqKkYqRiNGJ0YpLCYqJEYtRiNGI0YjRiNGI0YvRiNGKQ==

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

What are J1(Y0) and J1(Y1) ? Let us first express J1(yt)/t in terms of Z and U(y) We need to express Q(0,0). We use the expression of Q(x,0) in terms of V(x). The value of V when x=0 Vx0:=solve(numer(xTV),V)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWeDBHRigqJiwmSSJUR0YoIiIiRjEhIiJGMSwmRjBGMSIiJEYxRjI3I0Yu

Q00Z:=factor(subs(V=Vx0,T=Z^2,QxZV))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVRMDBaR0YoLCQqLCIjSyIiIilJIlpHRigiIiRGMSwqKiRGMkYxRjEqJkY0RjEpRjMiIiNGMSEiIiomRjRGMUYzRjFGMUY0RjFGMSwmRjNGMUYxRjFGOiksJiokRjhGMUYxRjRGMUY0RjpGMTcjRi4=

We could have used Q(0,yt) instead solve(numer(yTU),U)

NiQqJClJIlRHNiIjIiIiIiIjRigsJEYjISIi

factor(subs(U=Z,QytZU)); factor(subs(U=-Z,QytZU))

LCQqLCIjSyIiIilJIlpHNiIiIiRGJSwqKiRGJkYlRiUqJkYpRiUpRiciIiNGJSEiIiomRilGJUYnRiVGJUYpRiVGJSwmRidGJUYlRiVGLyksJiokRi1GJUYlRilGJUYpRi9GJQ==

Error, numeric exception: division by zero

denom(QytZU)

Ki4sJkkiWkc2IiIiIkYmISIiRiYsJkYkRiZGJkYmRiYpLCYqJClGJCIiI0YmRiYiIiRGJkYuRiYsKiomSSJVR0YlRiZGJEYmRiZGK0YmRjFGJiomRi5GJkYkRiZGJ0YmLCZGMUYmRiRGJkYmLCpGMEYmRitGJ0YxRidGMkYnRiY=

The factor U+Z tells us that the right value is U=Z Now the expression of J1(yt)/t in terms of U(y) J1y

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

J1ytovertZU:=factor(subs(y=y*t,Q(0,t*y)=QytZU,Q(0,0)=Q00Z,y=yZU,T=Z^2,t=sqrt(t2Z), J1y/t))

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

The roots of Delta(y): two quadratic Catalan-ish series algY:=factor(subs(y=Y,numer(Delta)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdZR0YoKiYsKiomKUkiWUdGKCIiIyIiIkkidEdGKEY0RjQqKEYzRjRGMkY0RjVGNEY0RjIhIiJGNUY0RjQsKkYwRjRGNkY3RjJGN0Y1RjRGNDcjRi4=

algY0:=op(1,%); algY1:=op(2,%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMEdGKCwqKiYpSSJZR0YoIiIjIiIiSSJ0R0YoRjNGMyooRjJGM0YxRjNGNEYzRjNGMSEiIkY0RjM3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMUdGKCwqKiYpSSJZR0YoIiIjIiIiSSJ0R0YoRjNGMyooRjJGM0YxRjNGNEYzISIiRjFGNkY0RjM3I0Yu

Y0ser:=proc(n) op(algeqtoseries(algY0,t,Y,n,true)): end: Y1ser:=proc(n) op(algeqtoseries(algY1,t,Y,n,true)): end: Y0ser(6); Y1ser(6)

KzFJInRHNiIiIiJGJSIiI0YmIiImIiIkIiM5IiIlIiNVRiciJEsiIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIig=

KzFJInRHNiIiIiJGJSEiIyIiIyIiJiIiJCEjOSIiJSIjVUYoISRLIiIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIo

We have expressions for J1(yt), so let us define YY0=Y0/t, and similarly for Y1 algYY0:=factor(subs(Y=t*YY,algY0/t)); algYY1:=factor(subs(Y=t*YY,algY1/t))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdZWTBHRigsKiomKUkjWVlHRigiIiMiIiIpSSJ0R0YoRjJGM0YzKihGMkYzRjFGM0Y1RjNGM0YxISIiRjNGMzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdZWTFHRigsKiomKUkjWVlHRigiIiMiIiIpSSJ0R0YoRjJGM0YzKihGMkYzRjFGM0Y1RjMhIiJGMUY3RjNGMzcjRi4=

Which series U corresponds to YY0 and YY1 ? We first find an equation of degree 6 over Q(T,t) yTU

LCQqLCIjOyIiIilJIlRHNiIiIiNGJSwmKiQpSSJVR0YoRilGJSEiIkYnRiVGJSwoKiZGJ0YlRi1GJUYlKiZGKUYlRidGJUYuRi1GJUYuLCwqJkYnRiVGLEYlRi4qJEYmRiVGJSooIiIpRiVGJ0YlRi1GJUYlRitGJSomIiIqRiVGJ0YlRi5GLkYu

res:=factor(resultant(numer(yTU-YY),algYY0,YY)); factor(rem(%,numer(t^2-t2T),t)); algU0Tt:=collect(%,U,factor)

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

Does not factor by introducing sqrt(T) factor(subs(T=Z^2,%))

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

Let us check n:=20: U0ser:=series(subs(y=convert(Y0ser(n),polynom)/t,convert(User(n),polynom)),t,n)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZVMHNlckdGKCtHSSJ0R0YoIiIiIiIhIiIjRjIiIzsiIiUiJG8iIiInRjQiIigiJScpPiIiKSIkOyIiIioiJiE0RCIjNSIlS0MiIzYiJ00yTCIjNyImZ10lIiM4IihbLlwlIiM5IidHYXkiIzoiKXFISWlGMyIpJVtGSyIiIzwiKjVNJSp5KSIjPSIqW0s3PSMiIz4tSSJPR0YlNiNGMCIjPzcjRi4=

series(subs(U=%,T=Tser(n),algU0Tt),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

Eliminate T res:=factor(resultant(algU0Tt,numer(t2T-t^2),T)); nops(%)

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

IiIk

n:=10: series(subs(U=U0ser,op(2,res)),t,n+1); series(subs(U=U0ser,op(3,res)),t,n+1)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Ky9JInRHNiIiKC9WPiUiIiciKVtzNkMiIigiKUdkNVciIikiKiEzUUk4IiIqIionKm80SSMiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

U0 has degree 8 over Q(t) algU0:=collect(op(2,res),U,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdVMEdGKCw8KigpSSJ0R0YoIiIjIiIiKSwmKiYiIiVGM0YxRjNGM0YzRjNGMkYzKUkiVUdGKCIiKUYzRjMqKkYyRjNGNUYzLCwqJiIja0YzKUYxRjdGM0YzKiZGOkYzKUYxIiIkRjNGMyomIiInRjNGMEYzRjNGNiEiIkYzRkVGMylGOSIiKEYzRjMqJiwwKiYiJSc0JUYzKUYxRkRGM0YzKiYiJUM1RjMpRjEiIiZGM0YzKiYiJSUpPkYzRj9GM0YzKiYiI0tGM0ZBRjNGMyomIiRTIkYzRjBGM0YzKiYiIykpRjNGMUYzRjNGMkYzRjMpRjlGREYzRjMqJiwuKiYiJkslPUYzRk9GM0YzKiYiJi9uIkYzRj9GM0YzKiYiJmdIIkYzRkFGM0YzKiYiJTdDRjNGMEYzRkUqJiIlJWUiRjNGMUYzRkUiI2FGM0YzKUY5RlBGM0YzKipGYG9GMywmRjFGM0YzRjNGMywsKiYiJSVlJEYzRj9GM0YzKiZGX29GM0ZBRjNGRSomIiQ/JkYzRjBGM0ZFKiYiJDgjRjNGMUYzRjNGM0ZFRjMpRjlGN0YzRjMqJiwuKiYiJkNRIkYzRk9GM0YzKiYiJ0MjWyhGM0Y/RjNGRSomIidrKTMiRjNGQUYzRjMqJiInX0M5RjNGMEYzRjMqJiIma0skRjNGMUYzRkUiJCdbRkVGMylGOUZCRjNGMyomLCwqJiInUykqcEYzRj9GM0YzKiYiJzspPSRGM0ZBRjNGMyomIidLPEtGM0YwRjNGRSomIiYpW19GM0YxRjNGM0ZpcEYzRjMpRjlGMkYzRjMqJiwsKiYiJyFHTCNGM0Y/RjNGRSomIicrS2VGM0ZBRjNGRSomIidvJ2UkRjNGMEYzRjMqJiImPyRlRjNGMUYzRkUiJWU5RjNGM0Y5RjNGMyomIiZrOyJGM0Y/RjNGMyomIidvZEdGM0ZBRjNGMyomIiduRDtGM0YwRjNGRSomIiZnIkhGM0YxRjNGM0ZgckZFNyNGLg==

Same for Y1 n:=20: U1ser:=series(subs(y=convert(Y1ser(n),polynom)/t,convert(User(n),polynom)),t,n)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZVMXNlckdGKCtHSSJ0R0YoIiIiIiIhIiIjRjIiIzsiIiUiJG8iIiInISIlIiIoIiUnKT4iIikhJDsiIiIqIiYhNEQiIzUhJUtDIiM2IidNMkwiIzchJmddJSIjOCIoWy5cJSIjOSEnR2F5IiM6IilxSElpRjMhKSVbRksiIiM8Iio1TSUqeSkiIz0hKltLNz0jIiM+LUkiT0dGJTYjRjAiIz83I0Yu

U0ser;

K0dJInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbyIiIidGKSIiKCIlJyk+IiIpIiQ7IiIiKiImITREIiM1IiVLQyIjNiInTTJMIiM3IiZnXSUiIzgiKFsuXCUiIzkiJ0dheSIjOiIpcUhJaUYoIiklW0ZLIiIjPCIqNU0lKnkpIiM9IipbSzc9IyIjPi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM/

series(U0ser-subs(t=-t,U1ser),t,20)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

algU1:=subs(t=-t,algU0)

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

Equation for J1(Y0) J1ytovertZU

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

We will eliminate U and Z to determine the equation over Q(t) res1:=factor(resultant(numer(J1ytovertZU-J1Y0/t), algU0, U)): indets(%); nops(%%)

PCVJJUoxWTBHNiJJIlpHRiRJInRHRiQ=

IiZHKD0=

res2:=factor(resultant(res1,numer(t^2-t2Z),Z)): nops(res2)

IiIl

map(nops,[seq(op(i,res2),i=1..nops(res2))]); map(indets,[seq(op(i,res2),i=1..nops(res2))])

NyYiIiIiIiMiJUwlKkYk

NyY8IjwjSSJ0RzYiPCRJJUoxWTBHRiZGJUYn

Let us check J1y

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiRGK0YsRiwqJkYkRiYtRig2JEYqRipGJkYm

J1Y0ser:=proc(n) series(subs(Q(0,y)=subs(x=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),y=Y0ser(n),J1y),t,n+1): end: n:=65: series(subs(J1Y0=J1Y0ser(n), op(3,res2)),t,n+1); series(subs(J1Y0=J1Y0ser(n), op(1,op(4,res2))),t,n+1);

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI2s=

degree 8 algJ1Y0:=collect(op(1,op(4,res2)),J1Y0,factor)

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

J1Y1ser:=proc(n) series(subs(Q(0,y)=subs(x=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),y=Y1ser(n),J1y),t,n+1): end: J1Y0ser(10);J1Y1ser(10); series(%%+subs(t=-t,%),t,10)

KzlJInRHNiIhIiJGJSIiIyIiISIiIkYoRiZGJiIiJSIiJCIjN0YpIiNKIiImIiQvIiIiJyIkLCQiIigiJWs1IiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIio=

KzlJInRHNiIhIiJGJSEiIyIiISIiIkYoRiYiIiMiIiUiIiQhIzdGKiIjSiIiJiEkLyIiIiciJCwkIiIoISVrNSIiKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIq

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

J1(Y1) is obtained by setting t=-t in J1(Y0), and taking the opposite algJ1Y1:=subs(J1Y0=-J1Y1,t=-t,algJ1Y0)

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

n:=20: series(subs(J1Y1=J1Y1ser(n),algJ1Y1),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz4=

What we really need are the sum and product, which are odd (resp. even) functions of t series(J1Y1ser(n)+J1Y0ser(n),t,n); series(J1Y1ser(n)*J1Y0ser(n),t,n)

KzlJInRHNiIhIiMhIiIiIiMiIiIiIikiIiQiI2kiIiYiJC0nIiIoIiVzbCIiKiImQ3IoIiM2IidrMSYqIiM4IilfTDk3IiM6IiovUE9mIiIjPC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiM+

KzlJInRHNiIiIiIhIiMhIiciIiEhIzoiIiMhJDEiIiIlISQpKSoiIichJlEwIiIiKSEnOj03IiM1ISgtZFsiIiM3ISk2J0cpPSIjOSEqM1NmWCMiIzstSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjPQ==

res1:=factor(resultant(sum-J1Y0-J1Y1,algJ1Y1,J1Y1)): nops(res1)

IiRXIg==

res2:=factor(resultant(res1,algJ1Y0,J1Y0)): nops(res2)

IiIm

map(nops,[seq(op(i,res2),i=1..5)])

NyciIiIiIiMiIzsiI0MiJDQn

map(indets,[seq(op(i,res2),i=1..5)])

Nyc8IjwjSSJ0RzYiPCRJJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJkYlRidGJw==

n:=10:series(subs(sum=J1Y0ser(n)+J1Y1ser(n),op(3,res2)),t,n)

Ky9JInRHNiIiI0YiIiEhJFcqIiIjIiZ3PCMiIiUhJ0dkRiIiJyEnS293IiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM1

series(subs(sum=J1Y0ser(n)+J1Y1ser(n),op(4,res2)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

n:=50:series(subs(sum=J1Y0ser(n)+J1Y1ser(n),op(5,res2)),t,n)

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

degree 8 algsum:=collect(op(4,res2),sum,factor)

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

series(subs(sum=J1Y0ser(n)+J1Y1ser(n),algsum),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI1w=

In terms of Z res1:=factor(resultant(subs(sum=t*sumovert,algsum),numer(t^2-t2Z),t)): nops(res1)

IiIo

map(indets,[seq(op(i,res1),i=1..7)]); map(nops,[seq(op(i,res1),i=1..7)])

Nyk8IjwjSSJaRzYiRiRGJDwkRiVJKXN1bW92ZXJ0R0YmRidGJw==

NykiIiIiIiNGJEYkRiRGJEYk

n:=10:series(subs(sumovert=(J1Y0ser(n)+J1Y1ser(n))/t,Z=Zser(n),op(1,op(5,res1))),t,n); series(subs(sumovert=(J1Y0ser(n)+J1Y1ser(n))/t,Z=Zser(n),op(1,op(6,res1))),t,n); n:=20:series(subs(sumovert=(J1Y0ser(n)+J1Y1ser(n))/t,Z=Zser(n),op(1,op(7,res1))),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

Ky9JInRHNiIhJGMjISIjISQ3JiIiISElK2siIiMhJkcsKSIiJSEoI1JeNSIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

Ky1JInRHNiIiMjNddjZxImZhJyoiIiEiNCVRZzxCTiVRazUlIiIjIjZvTnBydkthUFs5IiIiJSI3Iz5vKjMpKTNGZzpYRSIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

algsumovertZ:=op(1,op(5,res1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1hbGdzdW1vdmVydFpHRigsOiomKUkiWkdGKCIiKCIiIkkpc3Vtb3ZlcnRHRihGM0YzKiYpRjEiIidGM0Y0RjMhIiIqJiIja0YzRjZGM0YzKigiIipGMylGMSIiJkYzRjRGM0YzKiYiJCM+RjNGPUYzRjMqKEY8RjMpRjEiIiVGM0Y0RjNGOComRjpGM0ZCRjNGOCooIiNGRjMpRjEiIiRGM0Y0RjNGMyomRjpGM0ZHRjNGMyooRkZGMylGMSIiI0YzRjRGM0Y4KihGRkYzRjFGM0Y0RjNGMyomRkZGM0Y0RjNGODcjRi4=

sumJY:=t*factor(solve(%,sumovert))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzdW1KWUdGKCwkKi4iI2siIiJJInRHRihGMSlJIlpHRigiIiRGMSwqKiRGM0YxRjEqJkY1RjEpRjQiIiNGMUYxRjQhIiJGMUYxRjEsJkY0RjFGMUY7RjspLCYqJEY5RjFGMUY1RjFGNUY7Rjs3I0Yu

Check n:=20: series(subs(Z=Zser(n), sumJY-J1Y0ser(n)-J1Y1ser(n)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzw=

Now the product res1:=factor(resultant(pro-J1Y0*J1Y1,algJ1Y1,J1Y1)): nops(res1)

IiNT

res2:=factor(resultant(res1,algJ1Y0,J1Y0)): nops(res2)

IiIk

map(nops,[seq(op(i,res2),i=1..nops(res2))])

NyUiI1YiJWY2IiNX

map(indets,[seq(op(i,res2),i=1..nops(res2))])

NyU8JEkkcHJvRzYiSSJ0R0YlRiNGIw==

n:=90:series(subs(pro=J1Y0ser(n)*J1Y1ser(n),op(1,res2)),t,n):nops(%); series(subs(pro=J1Y0ser(n)*J1Y1ser(n),op(2,res2)),t,n):nops(%); n:=90:series(subs(pro=J1Y0ser(n)*J1Y1ser(n),op(3,res2)),t,n): nops(%);

IiMjKg==

IiMjKg==

IiIj

Choose the right factor degree 8 algpro:=collect(op(3,res2),pro,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdwcm9HRigsNCooIioqWz91USIiIilJInRHRigiIzlGMSlJJHByb0dGKCIiKUYxRjEqKiIpVU00JylGMSlGMyIjN0YxLCYqJiIjS0YxKUYzIiIjRjFGMSIjNkYxRjEpRjYiIihGMSEiIioqIihCVmYiRjEpRjMiIzVGMSwoKiYiJVNBRjEpRjMiIiVGMUYxKiYiJHMpRjFGP0YxRkQiJHAmRjFGMSlGNiIiJ0YxRjEqKiImJG8+RjEpRjNGN0YxLCoqJiIndyRII0YxKUYzRlJGMUYxKiYiJyEzMSJGMUZMRjFGMSomIidDW15GMUY/RjFGMSImeE8jRkRGMSlGNiIiJkYxRjEqKiIlKD0jRjFGWUYxLCwqJiInU0ElKUYxRlVGMUYxKiYiKSM+JT0pKUYxRllGMUYxKiYiKG8kWzVGMUZMRjFGMSomIihZJGZQRjFGP0YxRkQiJmBbJ0YxRjEpRjZGTUYxRjEqKiIjIilGMUZMRjEsLiomIigzQXElRjFGR0YxRjEqJiIrXypwRWAnRjFGVUYxRkQqJiIrQ1VbZURGMUZZRjFGMSomIionKkcqPXBGMUZMRjFGRComIilVUSc+JUYxRj9GMUYxIidGJz4kRkRGMSlGNiIiJEYxRjEqKkZARjFGP0YxLDAqJiIpd0x0QEYxRjpGMUYxKiYiLSc0QyRbLThGMUZHRjFGRComIi03IlE4UDQlRjFGVUYxRkQqJiIsMyFmIjRGJEYxRllGMUZEKiYiLE9TVSNlOUYxRkxGMUYxKiYiKip5eVBRRjFGP0YxRkQiKF5ISyJGMUYxKUY2RkBGMUYxKiYsMiomIihTOWkjRjFGMkYxRjEqJiIscyUqW1MpUkYxRjpGMUZEKiYiLiM+VypSQ2MiRjFGR0YxRkQqJiIuay9iKHlkOkYxRlVGMUZEKiYiLTNBT3kzRUYxRllGMUYxKiYiLGMlPkRWNkYxRkxGMUZEKiYiKSFlSyhwRjFGP0YxRjEiJ1x3NkZERjFGNkYxRjEqJiwyKiYiJGMjRjEpRjNGQ0YxRjEqJiImKVtqRjFGWUYxRjEqJiIoRyhwUkYxKUYzRmpuRjFGMSomIig7a3IkRjFGTEYxRjEqJiInO1slKkYxKUYzRmdwRjFGMSomIiVTc0YxRj9GMUYxKiYiJnZjIkYxRjNGMUZEIiQnb0YxRjEsMkZbc0YxRl5zRkRGYHNGMUZjc0ZERmVzRjFGaHNGREZqc0ZERlx0RkRGMUYxNyNGLg==

In terms of Z res1:=factor(resultant(algpro,numer(t^2-t2Z),t)): nops(res1)

IiIk

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

NyU8JEkiWkc2IkkkcHJvR0YlRiNGIw==

NyUiIiNGI0Yj

n:=10:series(subs(pro=(J1Y0ser(n)*J1Y1ser(n)),Z=Zser(n),op(1,op(1,res1))),t,n); series(subs(pro=(J1Y0ser(n)*J1Y1ser(n)),Z=Zser(n),op(1,op(2,res1))),t,n); n:=20:series(subs(pro=(J1Y0ser(n)*J1Y1ser(n)),Z=Zser(n),op(1,op(3,res1))),t,n);

Ky9JInRHNiIiJEciISIjIiRjIyIiISIlK0siIiMiJmsrJSIiJSInJ3BEJiIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

Ky1JInRHNiIiNksjemMmPkxQd1NVJCIiISE4I2Y5ZyJSemsvYWVtIiIiIyE5V2A9KVtcSkQmZWRwPSIiJSI6W0tAI3BANnROLTAvRSIiJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

algproZ:=op(1,op(2,res1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShhbGdwcm9aR0YoLEIqJilJIlpHRigiIigiIiJJJHByb0dGKEYzRjMqJiIiJUYzRjBGMyEiIiomKUYxIiInRjNGNEYzRjcqJiIjR0YzRjlGM0Y3KigiIipGMylGMSIiJkYzRjRGM0YzKiYiIzdGM0Y/RjNGMyooRj5GMylGMUY2RjNGNEYzRjcqJiIjd0YzRkRGM0Y3KigiI0ZGMylGMSIiJEYzRjRGM0YzKiYiJCE9RjNGSUYzRjMqKEZIRjMpRjEiIiNGM0Y0RjNGNyomIiQ3I0YzRk5GM0Y3KihGSEYzRjFGM0Y0RjNGMyomRjZGM0YxRjNGMyomRkhGM0Y0RjNGN0Y2Rjc3I0Yu

proJY:=factor(solve(%,pro))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwcm9KWUdGKCwkKioiIiUiIiIsMiokKUkiWkdGKCIiKEYxRjEqJkY2RjEpRjUiIidGMUYxKiYiIiRGMSlGNSIiJkYxISIiKiYiIz5GMSlGNUYwRjFGMSomIiNYRjEpRjVGO0YxRj4qJiIjYEYxKUY1IiIjRjFGMUY1Rj5GMUYxRjEsJkY1RjFGMUY+Rj4pLCYqJEZHRjFGMUY7RjFGO0Y+RjE3I0Yu

Check n:=20: series(subs(Z=Zser(n), proJY-J1Y0ser(n)*J1Y1ser(n)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzs=

J1ytovertZU

LCQqNiIjSyIiIilJIlpHNiIiIiRGJSxWKiYpSSJVR0YoIiIlRiUpRiciIidGJUYlKigiIiNGJSlGLUYyRiUpRiciIilGJSEiIiokKUYnIiM1RiVGJSooRi5GJUYsRiUpRiciIiZGJUY2KihGNUYlKUYtRilGJUYvRiVGNiooRjVGJUYzRiUpRiciIihGJUYlKihGNUYlRi1GJUY0RiVGJSomRi5GJSlGJyIiKkYlRjYqKEY8RiVGLEYlKUYnRi5GJUYlKihGJEYlRj5GJUY7RiVGJSooIiNJRiVGM0YlRi9GJUYlKihGJEYlRi1GJUZARiVGNiomRilGJUY0RiVGNiooRi5GJUYsRiVGJkYlRiUqKCIjW0YlRj5GJUZHRiVGNiooIiNTRiVGM0YlRjtGJUY2KigiIyEpRiVGLUYlRi9GJUY2KiYiI09GJUZARiVGJSooRkVGJUYsRiUpRidGMkYlRjYqKCIkSSJGJUYzRiVGR0YlRiUqJiIjckYlRi9GJUYlKigiI0NGJUY+RiVGV0YlRiUqKCIkXyJGJUYtRiVGR0YlRjYqJkYpRiVGLEYlRiUqKEYyRiVGM0YlRldGJUYlKiYiI0ZGJUZHRiVGJUYlLCoqJkYtRiVGJ0YlRiUqJEZXRiVGJUYtRiUqJkYpRiVGJ0YlRjZGNiwmRi1GJUYnRiVGNiwqRl9vRiVGYG9GNkYtRjZGYW9GNkY2LCZGLUYlRidGNkY2LCZGJ0YlRiVGNkY2LCZGJ0YlRiVGJUY2KSwmRmBvRiVGKUYlRilGNkYl

Expression of (D(yt)+2/(3t^2y))^2 expr:=4/9/y^2/subs(y=y*t,Delta)*(J1ytovert^2-sumovert*J1ytovert+proovert2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoLCQqKiMiIiUiIioiIiIpSSJ5R0YoIiIjISIiLChGM0YzKigpLCYqJkkidEdGKEYzRjVGM0YzRjNGN0Y2RjMpLCZGPEYzRjNGM0Y2RjNGNEY3RjMqKEY2RjNGNUY3LCYqJilGPUY2RjNGNEYzRjNGM0YzRjNGN0Y3LCgqJClJKkoxeXRvdmVydEdGKEY2RjNGMyomRkdGM0kpc3Vtb3ZlcnRHRihGM0Y3SSpwcm9vdmVydDJHRihGM0YzRjM3I0Yu

exprUZ:=factor(subs(y=yTU, J1ytovert=J1ytovertZU,sumovert=sumJY/t,proovert2=proJY/t^2,t=sqrt(t2Z),T=Z^2,expr))

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

This is a square is Q(Z,U). Choose the correct sign now exprroot:=factor(simplify(sqrt(exprUZ),symbolic))

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

factor(subs(Z=sqrt(T),expand(numer(%)/Z)))

LCQqLCIkRyIiIiJJIlRHNiJGJSwsKiZGJkYlKUkiVUdGJyIiI0YlISIiKiQpRiZGLEYlRiUqKCIiJUYlRiZGJUYrRiVGJSokRipGJUYtKiYiIiRGJUYmRiVGLUYlLCxGKUYtRi5GJSooIiIpRiVGJkYlRitGJUYlRjJGJSomIiIqRiVGJkYlRi1GJSksKComRiZGJUYrRiVGJSomRixGJUYmRiVGLUYrRiVGLEYlRiU=

factor(subs(Z=sqrt(T),expand(denom(%%))))

LCQqLCIiJCIiIiwmSSJURzYiRiVGJSEiIkYlKSwmRidGJUYkRiVGJEYlLCYqJClJIlVHRigiIiNGJUYpRidGJUYlLDYqJilGJ0YwRiUpRi8iIiVGJUYlKihGMEYlKUYnRiRGJUYuRiVGKSokKUYnRjVGJUYlKigiIztGJUY3RiVGL0YlRiUqKCIjQ0YlRjNGJUYuRiVGKSokRjRGJUYpKiZGPUYlRjdGJUYpKigiI1tGJUYzRiVGL0YlRiUqKCIiJ0YlRidGJUYuRiVGKSomIiIqRiVGM0YlRilGJUYp

exprrootTU:=sqrt(T)*%%/%;

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

factor(subs(T=Z^2,exprrootTU^2)/exprUZ)

IiIi

Let us check n:=20: factor(series(subs(U=User(n), T=Tser(n), exprrootTU),t,n+1))

Ky9JInRHNiIsJComIyIiIyIiJCIiIkkieUdGJCEiIkYqISIjRiciIiEsJiomRidGKkYrRipGKiMiIilGKUYqRigsKComRidGKilGK0YoRipGKiomIyIjP0YpRipGK0YqRiojIiNpRilGKiIiJSwqKiZGJ0YqKUYrRilGKkYqKiYiIzdGKkY1RipGKiomIyIkNyNGKUYqRitGKkYqIyIkLSdGKUYqIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGMg==

factor(series(subs(y=y*t,Dser(n))+2/3/t^2/y,t,8))

Ky9JInRHNiIsJComIyIiIyIiJCIiIkkieUdGJCEiIkYqISIjRiciIiEsJiomRidGKkYrRipGKiMiIilGKUYqRigsKComRidGKilGK0YoRipGKiomIyIjP0YpRipGK0YqRiojIiNpRilGKiIiJSwqKiZGJ0YqKUYrRilGKkYqKiYiIzdGKkY1RipGKiomIyIkNyNGKUYqRitGKkYqIyIkLSdGKUYqIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGMg==

factor(series(%%-%,t,8))

IiIh

Let us check that the degree of D(yt) is 24 res1:=factor(resultant(numer(exprroot-S),numer(y-yZU),U)): nops(res1)

IiIn

map(indets, [seq(op(i,res1),i=1..nops(res1))])

Nyg8IjwjSSJaRzYiRiRGJDwlSSJTR0YmRiVJInlHRiZGJA==

nops(op(5,res1))

IiRCIg==

res2:=factor(resultant(op(5,res1),subs(s=t^2, T=Z^2,algTs),Z)): nops(res2)

IiVcPA==

indets(res2)

PCVJIlNHNiJJInRHRiRJInlHRiQ=

degree(res2,S)

IiND

We need to express I1(x) I1x

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKEYoRiZGJCEiIkYnRi5GLg==

QxZV

LCQqNiIjOyIiIkkiVkc2IkYlKUkiWkdGJyIiJEYlLCoqJkYmRiUpRikiIiNGJUYlKiRGLUYlISIiRiZGMEYqRjBGJSxKKiYpRiZGKkYlKUYpIiIlRiVGJSooRi5GJUYzRiVGKEYlRjAqJilGJkYuRiVGNEYlRiUqKEY1RiVGM0YlRi1GJUYlKihGLkYlRjhGJUYoRiVGJSomRiZGJUY0RiVGJSooIiInRiVGM0YlRilGJUYwKigiIilGJUY4RiVGLUYlRjAqKEYuRiVGJkYlRihGJUYwKiRGNEYlRiUqJkYqRiVGM0YlRiUqKEYuRiVGOEYlRilGJUYwKihGP0YlRiZGJUYtRiVGMComRi5GJUYoRiVGJSomIiIoRiVGOEYlRiUqKEYuRiVGJkYlRilGJUYlKiZGNUYlRi1GJUYlKiZGR0YlRiZGJUYlKiZGPUYlRilGJUYlRipGJUYlKSwqKiZGJkYlRilGJUYlRiZGMEYpRjBGJUYwRi5GMCksJkYmRiVGJUYlRi5GMCksJkYvRiVGKkYlRipGMCwmRilGJUYlRiVGMCwmRilGJUYlRjBGMEYw

I1overtZV:=factor(subs(Q(x,0)= QxZV, t=sqrt(t2Z),x=xZV,I1x/t))

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

Expression of (Aminus(sqrt(x))+...)^2 expr:=x^2/9/(1+x)^2*(I1overt^2-sumovert*I1overt+proovert2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoLCQqKiMiIiIiIipGMSlJInhHRigiIiNGMSksJkY0RjFGMUYxRjUhIiIsKCokKUkoSTFvdmVydEdGKEY1RjFGMSomRjxGMUkpc3Vtb3ZlcnRHRihGMUY4SSpwcm9vdmVydDJHRihGMUYxRjE3I0Yu

exprVZ:=factor(subs(x=xZV, I1overt=I1overtZV,sumovert=sumJY/t,proovert2=proJY/t^2,T=Z^2,t=sqrt(t2Z),expr))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdleHByVlpHRigsJCo4IyIkYyMiIioiIiIpLEoqJilJIlZHRigiIiVGMylJIlpHRigiIidGM0YzKihGOUYzKUY4IiIkRjNGOkYzISIiKiZGN0YzKUY7RjlGM0YzKihGPEYzKUY4IiIjRjNGOkYzRjMqKCIjN0YzRj5GM0ZCRjNGQCooRjlGM0Y4RjNGOkYzRkAqKCIiJkYzRjdGMylGO0ZFRjNGQCooRjxGM0ZERjNGQkYzRjMqJEY6RjNGMyooRjlGM0Y+RjNGS0YzRjMqKEZHRjNGOEYzRkJGM0ZAKiZGP0YzRjdGM0YzKigiI01GM0ZERjNGS0YzRjMqJEZCRjNGMyomRkdGM0Y+RjNGMyooRjlGM0Y4RjNGS0YzRjMqJiIjPUYzRkRGM0YzKiZGSkYzRktGM0ZAKiZGR0YzRjhGM0YzRj9GM0ZFRjNGOkYzKSwqKiZGOEYzRktGM0YzKiRGS0YzRkAqJkY/RjNGOEYzRjNGM0YzRkVGM0ZERkApLCZGO0YzRjNGQEZFRkApLCoqJkY4RjNGO0YzRjNGOEZARjtGQEYzRkBGOUZAKSwqRl1vRjNGOEYzRjtGQEYzRjNGOUZAKSwmRjtGM0YzRjNGRUZAKSwmRmduRjNGP0YzRjxGQCksJkY4RjNGM0YzRkVGQEYzNyNGLg==

indets(%)

PCRJIlZHNiJJIlpHRiQ=

Again, this is a square in Q(Z,U) (choose the correct sign) exprrootZV:=factor(simplify(sqrt(exprVZ),symbolic))

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

factor(subs(Z=sqrt(T),expand(numer(%))))

LCQqKiIjOyIiIilJIlRHNiIjIiIkIiIjRiUsKiomRidGJUkiVkdGKEYlRiVGJyEiIiomRipGJUYuRiVGJUYlRiVGJSxKKiYpRidGKkYlKUYuIiIlRiVGJSooRjVGJUYzRiUpRi5GKkYlRi8qJilGJ0YrRiVGNEYlRiUqKCIiJ0YlRjNGJSlGLkYrRiVGJSooIiM3RiVGOUYlRjdGJUYvKigiIiZGJUYnRiVGNEYlRi8qKEY1RiVGM0YlRi5GJUYvKihGO0YlRjlGJUY8RiVGJSooRjVGJUYnRiVGN0YlRiUqJkYqRiVGNEYlRiUqJEYzRiVGJSooRj5GJUY5RiVGLkYlRi8qKCIjTUYlRidGJUY8RiVGJSomRj5GJUY3RiVGJSokRjlGJUYlKihGNUYlRidGJUYuRiVGJSomIiM9RiVGPEYlRiUqJkZARiVGJ0YlRi8qJkY+RiVGLkYlRiVGKkYlRiVGJQ==

factor(subs(Z=sqrt(T),expand(denom(%%))))

LCQqLiIiJCIiIkkiVkc2IkYlLCZGJkYlRiVGJUYlLCZJIlRHRidGJUYlISIiRiUpLCZGKkYlRiRGJUYkRiUpLC4qJkYqRiUpRiYiIiNGJUYlKihGMkYlRipGJUYmRiVGKyokRjFGJUYrRipGJSomRjJGJUYmRiVGK0YlRitGMkYlRiU=

exprrootTV:=%%/%

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

Let us check n:=20: factor(series(subs(V=Vser(n), T=Tser(n), exprrootTV),t,n+1))

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

factor(series(subs(x=1/sqrt(x),Amser(30))+x/2/t/(1+x)*(t*subs(y=0,Dser(n))-2/3/t/(1+x)),t,10))

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

factor(series(%%-%,t,10))

IiIh

Let us check that the degree of Aminus(sqrt(x)) is 24 res1:=factor(resultant(numer(exprrootZV-S),numer(x-xZV),V)): nops(res1)

IiIm

map(indets, [seq(op(i,res1),i=1..nops(res1))])

Nyc8IjwjSSJaRzYiRiRGJDwlSSJTR0YmRiVJInhHRiY=

nops(op(5,res1))

IiRmIg==

res2:=factor(resultant(op(5,res1),subs(s=t^2, T=Z^2,algTs),Z)): nops(res2)

IiU2Rg==

indets(res2)

PCVJIlNHNiJJInRHRiRJInhHRiQ=

degree(res2,S)

IiND

We denote s=t^2 with(plots):implicitplot(t2T-s,s=0..1/16,T=0..3)

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algTs:=numer(t2T-s)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdUc0dGKCwsKiQpSSJUR0YoIiIlIiIiRjMqKCIkYyNGMylGMSIiJEYzSSJzR0YoRjMhIiIqJiIiKUYzRjZGM0YzKiYiIz1GMylGMSIiI0YzRjMiI0ZGOTcjRi4=

factor(discrim(%,T))

LCQqKCItR0QlZkFJJCIiIilJInNHNiIiIiNGJSksJiomIiM7RiVGJ0YlRiVGJSEiIkYpRiVGLg==

map(simplify,map(allvalues,algeqtoseries(subs(s=1/16*(1-epsilon),algTs),epsilon,T,10))): Tsing:=op(3,%)

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Zsing:=series(sqrt(Tsing),epsilon,10)

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yTU

LCQqLCIjOyIiIilJIlRHNiIiIiNGJSwmKiQpSSJVR0YoRilGJSEiIkYnRiVGJSwoKiZGJ0YlRi1GJUYlKiZGKUYlRidGJUYuRi1GJUYuLCwqJkYnRiVGLEYlRi4qJEYmRiVGJSooIiIpRiVGJ0YlRi1GJUYlRitGJSomIiIqRiVGJ0YlRi5GLkYu

User(10)

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algUTy:=collect(numer(y-yTU),U,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdVVHlHRigsKioqSSJ5R0YoIiIiLCZJIlRHRihGMUYxISIiRjEsJkYzRjFGMUYxRjEpSSJVR0YoIiIkRjFGNCoqIiIjRjFGM0YxLCgqKCIiJkYxRjNGMUYwRjFGMSomIiIpRjFGM0YxRjQqJkY4RjFGMEYxRjFGMSlGN0Y6RjFGMSoqRjNGMUYwRjEsKCokKUYzRjpGMUYxKiYiI0NGMUYzRjFGNCIiKkY0RjFGN0YxRjEqKEY6RjFGRUYxLCgqJkYzRjFGMEYxRjFGPkY0KiZGSEYxRjBGMUY0RjFGNDcjRi4=

UserTm1(5)

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Uyser:=proc(n) op(1,simplify(map(allvalues,algeqtoseries(subs(T=Z^2,algUTy),y,U,n,true)),symbolic)): end: Uyser(5)

Ky9JInlHNiJJIlpHRiQiIiEsJCooIyIiIiIiJUYqKSwmRiVGKkYqISIiIiIkRiopRiUiIiNGLkYqRiosJCoqI0YqIiNrRiosKiokKUYlRi9GKkYqKiYiIihGKkYwRipGLkYlRi5GKkYuRiopRi1GK0YqKUYlIiImRi5GLkYxLCQqKiNGKiIlQzVGKiwwKiQpRiUiIidGKkYqKiYiIz1GKkY8RipGLiomIiNiRiopRiVGK0YqRioqJiIjP0YqRjhGKkYqKiYiIzpGKkYwRipGKiomRjFGKkYlRipGLkY6Ri5GKilGLUY9RiopRiUiIilGLkYqRi8sJCoqI0YqIiYlUTtGKiw2KiQpRiUiIipGKkYqKiYiI0xGKkZRRipGLiomIiRfI0YqKUYlRjpGKkYqKiYiJEclRipGREYqRi4qJiIkI0dGKkY8RipGLiomIiRZI0YqRkpGKkYuKiZGK0YqRjhGKkYuKiYiJCE9RipGMEYqRioqJkZmbkYqRiVGKkYqRk5GKkYqKUYtRkVGKilGJSIjNkYuRi5GKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRj0=

Q(0,yt) is a series in y with coefficients in Q(Z) Qytsery:=proc(n) factor(series(subs(U=Uyser(n),QytZU),y,n+1)): end: Qytsery(4);

Ky1JInlHNiIsJCosIiNLIiIiKUkiWkdGJCIiJEYoLCoqJEYpRihGKComRitGKClGKiIiI0YoISIiKiZGK0YoRipGKEYoRitGKEYoLCZGKkYoRihGKEYxKSwmKiRGL0YoRihGK0YoRitGMUYoIiIhKiosJkYqRihGKEYxRihGM0YoLCgqJClGKiIiJUYoRigqJiIiJ0YoRi9GKEYxIiNARihGKEY0RjFGKCwkKixGMEYoRilGMSlGOUYwRigsLiokKUYqRj9GKEYoKiZGMEYoKUYqIiImRihGMUY7RjEqJiIiKEYoRi9GKEYoKiZGMEYoRipGKEYoRihGKEYoRjRGMUYoRjAsJCosI0YoIiM7RigpRjlGK0YoRkZGMSw2KiZGK0YoKUYqIiIqRihGKComIiNCRigpRioiIilGKEYxKiYiI0VGKClGKkZLRihGKComIiNBRihGRkYoRigqJiIjV0YoRkhGKEYoKiYiJCsiRihGPEYoRjEqJiIjIylGKEYpRihGMSomIiNJRihGL0YoRjEqJkZVRihGKkYoRihGK0YoRihGNEYxRjFGKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRj0=

n:=20:series(subs(Z=Zser(n), coeff(%%,y,3)/t^3),t,n)

KzNJInRHNiIiIiIiIiQiIzkiIiYiJHMiIiIoIiUnNCMiIioiJkZmIyIjNiInPHBLIiM4IigiZSo+JSIjOi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM8

coeff(coeff(Qser(17),y,3),x,0)

LDIqJiIpclknWyYiIiIpSSJ0RzYiIiM8RiVGJSomIigiZSo+JUYlKUYnIiM6RiVGJSomIic8cEtGJSlGJyIjOEYlRiUqJiImRmYjRiUpRiciIzZGJUYlKiYiJSc0I0YlKUYnIiIqRiVGJSomIiRzIkYlKUYnIiIoRiVGJSomIiM5RiUpRiciIiZGJUYlKiQpRiciIiRGJUYl

Singular behaviour of U: There occurs a cubic function of y that we call M(y) Using:=map(factor,subs(eta=epsilon^(1/3), RootOf(2*y*_Z^3+(-27*y+36)*_Z^2+54*y*_Z-27*y-108)=M(y), op(algeqtoseries(subs(T=simplify(subs(epsilon=eta^3,convert(Tsing,polynom)),symbolic),algUTy),eta,U,5))))

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

alias(Mexpl=RootOf(2*y*_Z^3+(-27*y+36)*_Z^2+54*y*_Z-27*y-108))

SSZNZXhwbEc2Ig==

algM:=subs(_Z=M,2*y*_Z^3+(-27*y+36)*_Z^2+54*y*_Z-27*y-108)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdNR0YoLCwqKCIiIyIiIkkieUdGKEYxKUkiTUdGKCIiJEYxRjEqJiwmKiYiI0ZGMUYyRjEhIiIiI09GMUYxKUY0RjBGMUYxKigiI2FGMUYyRjFGNEYxRjFGOEY6IiQzIkY6NyNGLg==

yM:=factor(solve(algM,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5TUdGKCwkKioiI08iIiIsJiokKUkiTUdGKCIiI0YxRjEiIiQhIiJGMSwmKiZGNkYxRjVGMUYxRjdGOEY4LChGM0YxKiYiIzdGMUY1RjFGOCIiKkYxRjhGODcjRi4=

Let us choose the right series M(y): it is the value of U at t=1/4 Msersol:=map(allvalues,algeqtoseries(algM,y,M,3))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShNc2Vyc29sR0YoNyUrK0kieUdGKCEjPSEiIiMiI0YiIiMiIiEjIiImIiIkIiIiLUkiT0dGJTYjRjpGNSsrRjAqJClGOSNGOkY1RjpGNiwmI0Y4IiInRjIqJkZBRjpGQEY6RjpGOiwmIyIjPCIjQ0YyKiYjIiMqKSIkOyNGOkZARjpGOkY1RjtGOSsrRjAsJEY/RjJGNiwmRkNGMkZFRjJGOiwmRkdGMkZKRjJGNUY7Rjk3I0Yu

User(8)

Ky9JInRHNiIiIiIiIiEiIiNGJyIjOyIiJSwmIiRtIkYlKiZGJ0YlSSJ5R0YkRiVGJSIiJywoKiZGJ0YlKUYtRidGJUYlKiYiI1NGJUYtRiVGJSIlTT5GJSIiKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM1

Mser:=proc(n) op(1,map(allvalues,algeqtoseries(algM,y,M,n,true))): end: Mser(5)

Ky9JInlHNiIqJCkiIiQjIiIiIiIjRikiIiEsJiMiIiYiIichIiIqJkYoRilGJkYpRilGKSwmIyIjPCIjQ0YwKiYjIiMqKSIkOyNGKUYmRilGKUYqLCYjIiVmPCIlI2YjRjAqJiMiJDgiIiQpR0YpRiZGKUYpRicsJiomIyImLEYiIiYvNiRGKUYmRilGKSMiJUp0IiZvLiJGMCIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YpRi4=

We can simplify the expansion of U using the expression of y in terms of M Using:=map(factor,subs(M(y)=M,y=yM,Using))

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

Singular behaviour of V algVTx:=numer(x-xTV)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdWVHhHRigsMiooSSJUR0YoIiIiKUkiVkdGKCIiJEYxSSJ4R0YoRjFGMSooRjBGMSlGMyIiI0YxRjVGMSEiIiomRjJGMUY1RjFGOSooRjRGMUY3RjFGNUYxRjkqJkYwRjFGM0YxRjFGMEY5KiZGNEYxRjNGMUYxRjFGMTcjRi4=

This is where we introduce the series P(x) algeqtoseries(subs(T=simplify(subs(epsilon=eta^3,convert(Tsing,polynom)),symbolic),algVTx),eta,V,2)

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

Vsing:= map(factor,subs(eta=epsilon^(1/3), RootOf(_Z^3*x-3*_Z^2*x+3*_Z-1)=P(x), op(algeqtoseries(subs(T=simplify(subs(epsilon=eta^3,convert(Tsing,polynom)),symbolic),algVTx),eta,V,5) )))

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

alias(Pexpl=RootOf(_Z^3*x-3*_Z^2*x+3*_Z-1))

NiRJJk1leHBsRzYiSSZQZXhwbEdGJA==

algP:=subs(_Z=P,_Z^3*x-3*_Z^2*x+3*_Z-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdQR0YoLCoqJilJIlBHRigiIiQiIiJJInhHRihGM0YzKihGMkYzKUYxIiIjRjNGNEYzISIiKiZGMkYzRjFGM0YzRjNGODcjRi4=

xP:=factor(solve(algP,x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4UEdGKCwkKigsJiomIiIkIiIiSSJQR0YoRjNGM0YzISIiRjMpRjQiIiNGNSwmRjRGM0YyRjVGNUY1NyNGLg==

map(allvalues,algeqtoseries(algP,x,P,3))

NyUrK0kieEc2IiMiIiIiIiQiIiEjIiIpIiMiKUYnIyIjUyIkSCgiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJ0YoLCoqKF4jRidGJylGKCNGJ0YwRicpRiRGOyEiIkYnIyIiJUYoRicqKCwkKiYjRj8iIipGJ0Y5RidGJ0YnRjpGJ0Y8RidGPS1GMjYjRiRGJywqKigsJEY5RidGJ0Y6RidGPEY9Rj1GPkYnKihGQkYnRjpGJ0Y8RidGJ0ZFRic=

Choose the correct value of P(y) Vser(4)

KylJInRHNiIiIiIiIiMsJkkieEdGJEYlIiIpRiUiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiJw==

Pser:=proc(n) op(algeqtoseries(algP,x,P,n,true)): end: Pser(5)

Ky9JInhHNiIjIiIiIiIkIiIhIyIiKSIjIilGJiMiI1MiJEgoIiIjIyIkRygiJiRvPkYnIyImb1AlIihCVmYiIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIiY=

We can simplify the expansion of V using the expression of x in terms of P Vsing:=map(factor,subs(P(x)=P,x=xP,Vsing))

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

exprrootTU

LCQqNCMiJEciIiIkIiIiKUkiVEc2IiNGJiIiI0YnLCwqJkYpRicpSSJVR0YqRixGJyEiIiokKUYpRixGJ0YnKigiIiVGJ0YpRidGMEYnRicqJEYvRidGMSomRiZGJ0YpRidGMUYnLCxGLkYxRjJGJyooIiIpRidGKUYnRjBGJ0YnRjZGJyomIiIqRidGKUYnRjFGJyksKComRilGJ0YwRidGJyomRixGJ0YpRidGMUYwRidGLEYnLCZGKUYnRidGMUYxKSwmRilGJ0YmRidGJkYxLCZGNkYxRilGJ0YxLDYqJkYzRicpRjBGNUYnRicqKEYsRicpRilGJkYnRi9GJ0YxKiQpRilGNUYnRicqKCIjO0YnRklGJ0YwRidGJyooIiNDRidGM0YnRi9GJ0YxKiRGR0YnRjEqJkZPRidGSUYnRjEqKCIjW0YnRjNGJ0YwRidGJyooIiInRidGKUYnRi9GJ0YxKiZGPEYnRjNGJ0YxRjFGMQ==

n:=16: factor(series(subs(U=User(n),T=Tser(n),exprrootTU),t,n+1))

KytJInRHNiIsJComIyIiIyIiJCIiIkkieUdGJCEiIkYqISIjRiciIiEsJiomRidGKkYrRipGKiMiIilGKUYqRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiIiJQ==

exprsingM:=map(factor,series(subs(U=Using,T=Tsing, exprrootTU),epsilon,1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpleHByc2luZ01HRigsKCo2IyIjSyIjRiIiIikiIiQjRjMiIiNGMywoSSJNR0YoRjMiIichIiIqJkY1RjNGNEYzRjNGMywoRjlGM0Y6RjtGPEY7RjNGOUYzKSwmKiZGN0YzRjlGM0YzRjVGO0Y3RjMsJkY5RjMqJEY0RjNGO0Y7LCgqJClGOUY3RjNGMyomRjpGM0Y5RjNGMyIiKkY7RjssJkY5RjNGNUY7RjssJkY5RjNGQkYzRjtGMyo0IyIiJUZHRjMpSShlcHNpbG9uR0YoI0Y3RjVGM0Y+RjMpLChGREYzKiYiIzdGM0Y5RjNGO0ZHRjNGN0YzRjRGMylGNyNGM0Y1RjNGOUYzKUZIIiImRjtGQ0Y7RjMtSSJPR0YlNiNGTkYzNyNGLg==

b1:=(subs(epsilon=1,op(2,exprsingM)))

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evalf(GAMMA(-2/3))

JCErMHlTPVMhIio=

algb1:=factor(resultant(numer(bb1-b1),algM,M)); collect(%,bb1,factor)

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

LCgqKiIxc206Y0g6Xz4iIiIsKCokKUkieUc2IiIiI0YlRiUqJiIjQ0YlRilGJSEiIiIjO0YlRiUpLCZGKUYlIiIlRi4iIiZGJSlJJGJiMUdGKiIiJEYlRiUqKiI0c0VAVFArJ3lgTkYlKUYrI0YrRjZGJSlGMUY2RiVGNUYlRi4qKCI1S094U3NPNy5SbkYlKUY2I0YlRitGJSwmRilGJUYyRiVGJUYl

HyL:=-2^(5/3)*subs(M=L,b1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeUxHRigsJCowIyIjOyIiKiIiIiksJiomIiIjRjNJIkxHRihGM0YzIiIkISIiRjdGMyksKCokKUY4RjdGM0YzKiYiIzdGM0Y4RjNGOkYyRjNGN0YzKUY5I0YzRjdGM0Y4RjMpLCZGOEYzRjlGOiIiJkY6LChGPUYzKiYiIidGM0Y4RjNGM0YyRjpGOkY6NyNGLg==

algL:=subs(y=4*y,M=L,algM)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdMR0YoLCwqKCIiKSIiIkkieUdGKEYxKUkiTEdGKCIiJEYxRjEqJiwmKiYiJDMiRjFGMkYxISIiIiNPRjFGMSlGNCIiI0YxRjEqKCIkOyNGMUYyRjFGNEYxRjFGOEY6RjlGOjcjRi4=

yL:=factor(solve(algL,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5TEdGKCwkKioiIioiIiIsJiokKUkiTEdGKCIiI0YxRjEiIiQhIiJGMSwmKiZGNkYxRjVGMUYxRjdGOEY4LChGM0YxKiYiIzdGMUY1RjFGOEYwRjFGOEY4NyNGLg==

factor(HyL*yL^2/sqrt(3))

LCQqLCIkVyIiIiJJIkxHNiJGJSksJkYmRiUiIiQhIiIiIiZGKywoKiQpRiYiIiNGJUYlKiYiIidGJUYmRiVGJSIiKkYrRispLCZGLkYlRipGK0YwRiVGKw==

Lser:=proc(n) subs(y=4*y,Mser(n)): end

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVMc2VyR0YoZio2I0kibkdGKEYoRihGKC1JJXN1YnNHRiY2JC9JInlHRigsJComIiIlIiIiRjVGOUY5LUklTXNlckdGKEYvRihGKEYoNyNGLg==

Lser(4)

Ky0sJComIiIlIiIiSSJ5RzYiRiZGJiokKSIiJCNGJiIiI0YmIiIhLCYjIiImIiInISIiKiZGLEYmRipGJkYmRiYsJiMiIzwiI0NGMyomIyIjKikiJDsjRiZGKkYmRiZGLSwmIyIlZjwiJSNmI0YzKiYjIiQ4IiIkKUdGJkYqRiZGJkYrLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRiZGJQ==

Hyser:=proc(n) map(expand,map(rationalize,series(subs(L=Lser(n+2),HyL),y,n))) end

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIeXNlckdGKGYqNiNJIm5HRihGKEYoRigtSSRtYXBHRiY2JEknZXhwYW5kR0YmLUYyNiRJLHJhdGlvbmFsaXplR0YlLUknc2VyaWVzR0YmNiUtSSVzdWJzR0YmNiQvSSJMR0YoLUklTHNlckdGKDYjLCZGMCIiIiIiI0ZESSRIeUxHRihJInlHRihGMEYoRihGKDcjRi4=

Let us check N:=80:i:=0: plot([seq([1/n,coeff(coeff(convert(Dser(N),polynom),y,i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=1.2..1.7,style=point);

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Ig==

Hyser(2)

KylJInlHNiIsJComIyIjSyIiKiIiIikiIiQjRioiIiNGKkYqIiIhLCYjIiRHIkYpISIiKiZGMUYqRitGKkYqRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYu

evalf(-coeff(%,y,i)/GAMMA(-2/3))

JCIrMiFbRGAiISIq

N:=80:i:=1: plot([seq([1/n,coeff(coeff(convert(Dser(N),polynom),y,i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=1.5..3,style=point);

NistJSdDVVJWRVNHNiQ3XHA3JCQiIzUhIiIkIitubW1tOyEjNTckJCIiJiEiIiQiIiEhIiI3JCQiK0xMTExMISM1JCIrWj5FK2whIzU3JCQiI0QhIiMkIiIhISIiNyQkIiIjISIiJCIrNyVRKjM1ISIqNyQkIitubW1tOyEjNSQiIiEhIiI3JCQiK0g5ZEc5ISM1JCIrJTNpNkUiISIqNyQkIiREIiEiJCQiIiEhIiI3JCQiKzY2NjY2ISM1JCIqUHBaVyIhIik3JCQiIiIhIiIkIiIhISIiNyQkIisiNDQ0NCohIzYkIitWJkdPZSIhIio3JCQiK0xMTEwkKSEjNiQiIiEhIiI3JCQiKyNwMkJwKCEjNiQiKyQqWzQjcCIhIio3JCQiK1ZyJkc5KCEjNiQiIiEhIiI3JCQiK25tbW1tISM2JCIrPjk0ejwhIio3JCQiJEQnISIlJCIiISEiIjckJCIyMSsrNSVITiMpZSEjPSQiK3AiKVJdPSEiKjckJCIrY2JiYmIhIzYkIiIhISIiNyQkIismKnk6al8hIzYkIisobzAqND4hIio3JCQiIiYhIiMkIiIhISIiNyQkIitpWiE+dyUhIzYkIis4QEtnPiEiKjckJCIrWFhYWFghIzYkIiIhISIiNyQkIisoM0V5TSUhIzYkIisvISllLj8hIio3JCQiK25tbW1UISM2JCIiISEiIjckJCIiJSEiIyQiK1ciSDYvIyEiKjckJCIrWVE6WVEhIzYkIiIhISIiNyQkIisvUHEuUCEjNiQiK0FoLHU/ISIqNyQkIityJkc5ZCQhIzYkIiIhISIiNyQkIitpZUZbTSEjNiQiK0Qjb0k1IyEiKjckJCIrTExMTEwhIzYkIiIhISIiNyQkIitfayFlQSQhIzYkIiopSCMqR0AhIik3JCQiJURKISImJCIiISEiIjckJCIqLi4uLiQhIzUkIitvUTNfQCEiKjckJCIrcms8VEghIzYkIiIhISIiNyQkIitkRzlkRyEjNiQiKyJ5YEg8IyEiKjckJCIreXh4eEYhIzYkIiIhISIiNyQkIisuRnEtRiEjNiQiK1AqZT0+IyEiKjckJCIrWip5OmojISM2JCIiISEiIjckJCIra0Q1a0QhIzYkIissajE0QSEiKjckJCIjRCEiJCQiIiEhIiI3JCQiKlJDIVJDISM1JCIraGt6Q0EhIio3JCQiKyJRXzRRIyEjNiQiIiEhIiI3JCQiKyZSImVEQiEjNiQiK3dLQlJBISIqNyQkIit0c3NzQSEjNiQiIiEhIiI3JCQiK0FBQUFBISM2JCIrSjdgX0EhIio3JCQiK1ZJIlI8IyEjNiQiIiEhIiI3JCQiK3UmZnc3IyEjNiQiK3o1I1tFIyEiKjckJCIrTExMJDMjISM2JCIiISEiIjckJCIrRmoiMy8jISM2JCIrS1VAd0EhIio3JCQiIiMhIiMkIiIhISIiNyQkIis5VnlnPiEjNiQiKkExb0cjISIpNyQkIitCcDJCPiEjNiQiIiEhIiI3JCQiK2BDeicpPSEjNiQiK0Qlem1IIyEiKjckJCIrXz0mPSY9ISM2JCIiISEiIjckJCIrPT09PT0hIzYkIituXyFmSSMhIio3JCQiKydHOWR5IiEjNiQiIiEhIiI3JCQiK2xmUWE8ISM2JCIreGZhOUIhIio3JCQiK0p6OEM8ISM2JCIiISEiIjckJCIrYV8iXHAiISM2JCIrRWdsQUIhIio3JCQiK25tbW07ISM2JCIiISEiIjckJCIraVVNUjshIzYkIiszTEdJQiEiKjckJCIrRUshSGgiISM2JCIiISEiIjckJCIrKGUsdGUiISM2JCIrJTRxdUwjISIqNyQkIiZEYyIhIickIiIhISIiNyQkIitROllROiEjNiQiK1lRRFdCISIqNyQkIis6Ojo6OiEjNiQiIiEhIiI3JCQiKzh0YCNcIiEjNiQiK3R5bV1CISIqNyQkIitOIyllcTkhIzYkIiIhISIiNyQkIitpYEZcOSEjNiQiKzE+dWNCISIqNyQkIitIOWRHOSEjNiQiIiEhIiI3JCQiKy8yWDM5ISM2JCIrW0RdaUIhIio3JCQiKyopKSkpKSlRIiEjNiQiIiEhIiI3JCQiKzlJJylwOCEjNiQiKyZwdHpPIyEiKjckJCIrXjhOXjghIzYkIiIhISIiNyQkIitMTExMOCEjNiQiK2NvPHRCISIqNyQkIit1JSp5OjghIzYkIiIhISIiNyQkIisqSCwoKUgiISM2JCIrZjk4eUIhIio3JCQiKyNHXj9HIiEjNiQiIiEhIiI3JCQiKyZ5QWVFIiEjNiQiKyYzYkdRIyEiKjckJCIkRCIhIiUkIiIhISIiLSUmQ09MT1JHNiclJFJHQkckIilDKWVxJSEiKSQiIiEhIiIkIignPiFcJiEiKS0lK19BVFRSSUJVVEVHNiMvJSdzb3VyY2VHJSxtYXRoZGVmYXVsdEctJSVWSUVXRzYkOyQiIiEhIiIkIiImISIjOyQiIzohIiIkIiNJISIiLSYlJl9BWElTRzYjIiIiNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJiUmX0FYSVNHNiMiIiM2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lK0FYRVNMQUJFTFNHNiQtSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USRhYnM2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJmZhbHNlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiItSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USRvcmQ2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJlNUWUxFRzYjJSZQT0lOVEctJSlfVklTSUJMRUc2IyIiIi0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJFMoISIiLSUpQk9VTkRTX1lHNiMkIiRTIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJVNJISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJT9NISIiLSUpQ0hJTERSRU5HNiItJStBTk5PVEFUSU9ORzYnLSUpQk9VTkRTX1hHNiMkIiIhISIiLSUpQk9VTkRTX1lHNiMkIiIhISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK1MhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK1MhIiItJSlDSElMRFJFTkc2Ig==Ig==

Hyser(2)

KylJInlHNiIsJComIyIjSyIiKiIiIikiIiQjRioiIiNGKkYqIiIhLCYjIiRHIkYpISIiKiZGMUYqRitGKkYqRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYu

evalf(-coeff(%,y,i)/GAMMA(-2/3))

JCIrOFMjNGYjISIq

N:=100:i:=2: plot([seq([1/n,coeff(coeff(convert(Dser(N),polynom),y,i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=2.5..3.5,style=point);

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NiI=

Hyser(3)

KytJInlHNiIsJComIyIjSyIiKiIiIikiIiQjRioiIiNGKkYqIiIhLCYjIiRHIkYpISIiKiZGMUYqRitGKkYqRiosJiomIyIlU1kiIyIpRipGK0YqRiojIiRjI0YsRjNGLi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRiw=

evalf(-coeff(%,y,2)/GAMMA(-2/3))

JCIrSUpXYk0hIio=

H00:=coeff(Hyser(2),y,0)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIMDBHRigsJComIyIjSyIiKiIiIikiIiQjRjMiIiNGM0YzNyNGLg==

Expression of (Aminus(sqrt(x))+... exprrootTV

LCQqNCMiIzsiIiQiIiIpSSJURzYiI0YmIiIjRicsKiomRilGJ0kiVkdGKkYnRidGKSEiIiomRiZGJ0YvRidGJ0YnRidGJyxKKiYpRilGJkYnKUYvIiIlRidGJyooRjZGJ0Y0RicpRi9GJkYnRjAqJilGKUYsRidGNUYnRicqKCIiJ0YnRjRGJylGL0YsRidGJyooIiM3RidGOkYnRjhGJ0YwKigiIiZGJ0YpRidGNUYnRjAqKEY2RidGNEYnRi9GJ0YwKihGPEYnRjpGJ0Y9RidGJyooRjZGJ0YpRidGOEYnRicqJkYmRidGNUYnRicqJEY0RidGJyooRj9GJ0Y6RidGL0YnRjAqKCIjTUYnRilGJ0Y9RidGJyomRj9GJ0Y4RidGJyokRjpGJ0YnKihGNkYnRilGJ0YvRidGJyomIiM9RidGPUYnRicqJkZBRidGKUYnRjAqJkY/RidGL0YnRidGJkYnRidGL0YwLCZGL0YnRidGJ0YwLCZGKUYnRidGMEYwKSwmRilGJ0YmRidGJkYwKSwuKiZGKUYnRj1GJ0YnKihGLEYnRilGJ0YvRidGMCokRj1GJ0YwRilGJyomRixGJ0YvRidGMEYnRjBGLEYwRic=

n:=16: factor(series(subs(V=Vser(n),T=Tser(n),exprrootTV),t,n+1))

Ky1JInRHNiIsJCooIyIiIiIiJEYoSSJ4R0YkRigpLCZGKkYoRihGKCIiIyEiIkYuISIjLCQqKkYnRihGKkYoLCZGKkYoRi1GKEYoRixGLkYoIiIhLCQqKkYnRihGKkYoLCgqJClGKkYtRihGKComIiImRihGKkYoRigiIilGKEYoRixGLkYoRi0sJCoqRidGKEYqRigsKiomRi1GKClGKkYpRihGKComIiM5RihGOEYoRigqJiIjVkYoRipGKEYoIiNpRihGKEYsRi5GKCIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIn

exprsingP:=map(factor,series(subs(V=Vsing,T=Tsing, exprrootTV),epsilon,2))

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

b1:=factor(subs(epsilon=1,op(2,exprsingP)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNiMUdGKCwkKjQjIiIlIiIqIiIiLCYqJiIiJEYzSSJQR0YoRjNGM0YzISIiRjMsKCokKUY3IiIjRjNGMyomIiInRjNGN0YzRjhGM0YzRjMpRjYjRjNGPEYzKUY8I0YzRjZGMywoRjdGOEY8RjMqJEY/RjNGM0Y4LChGN0YzRkRGM0Y8RjhGOCwmRjdGM0YzRjNGOCksJkY3RjNGM0Y4RjxGOEYzNyNGLg==

evalf(GAMMA(-2/3))

JCErMHlTPVMhIio=

algb1:=factor(resultant(numer(bb1-b1),algP,P)); collect(%,bb1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdiMUdGKCw8KigiJndYIyIiIikiIiQjRjEiIiNGMSlJInhHRigiIiZGMSEiIiooIidjdTlGMUYyRjEpRjciIiVGMUY5KioiJnM5JUYxKUY1I0Y1RjNGMUkkYmIxR0YoRjFGNkYxRjEqKEYwRjFGMkYxKUY3RjNGMUY5KipGP0YxRkBGMUZCRjFGPEYxRjkqKkY/RjFGQEYxRkJGMUZERjFGOSooIiZHTCNGMSlGQkYzRjFGNkYxRjkqKkY/RjFGQEYxRkJGMSlGN0Y1RjFGMSooRkhGMUZJRjFGPEYxRjkqKCImY20lRjFGSUYxRkRGMUYxKihGTkYxRklGMUZLRjFGMSooRkhGMUZJRjFGN0YxRjkqJkZIRjFGSUYxRjk3I0Yu

LCgqKiImR0wjIiIiKSwmSSJ4RzYiRiVGJSEiIiIiI0YlKSwmRihGJUYlRiUiIiRGJSlJJGJiMUdGKUYuRiVGKiouIiZzOSVGJSlGKyNGK0YuRiVGLUYlKUYoRitGJUYmRiVGMEYlRiUqKiImd1gjRiUpRi4jRiVGK0YlKUYoRi5GJSwoKiRGNUYlRiUqJiIiJ0YlRihGJUYlRiVGJUYlRio=

Mind the term D0 in the expression of Aminus: it leads us to add another part to b1 HxP:=-2^(5/3)*b1-x/2/(1+x)*H00

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeFBHRigsJioyIyIjOyIiKiIiIiwmKiYiIiRGM0kiUEdGKEYzRjNGMyEiIkYzLCgqJClGNyIiI0YzRjMqJiIiJ0YzRjdGM0Y4RjNGM0YzKUY2I0YzRjxGMywoRjdGOEY8RjMqJEY/RjNGM0Y4LChGN0YzRkJGM0Y8RjhGOCwmRjdGM0YzRjNGOCksJkY3RjNGM0Y4RjxGOEY4KipGMEYzSSJ4R0YoRjMsJkZIRjNGM0YzRjhGP0YzRjg3I0Yu

HxP:=factor(subs(x=xP,HxP))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeFBHRigsJCosIyIjSyIiKiIiIiwmKiYiIiRGM0kiUEdGKEYzRjNGMyEiIkYzKUY2I0YzIiIjRjMsJkY3RjNGM0YzRjgpLCZGN0YzRjNGOEY7RjhGMzcjRi4=

Let us check Cmser(4)

LCYqJiwmKiYiIioiIiIpSSJ0RzYiIiIlRidGJyokKUYpIiIjRidGJ0YnKUkieEdGKkYuISIiRicqKEYuRicpRjBGK0YxRihGJ0Yn

N:=60:i:=1: plot([seq([1/n,coeff(coeff(Cmser(N),x,-2*i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.1,ord=0.5..0.8,style=point);

NistJSdDVVJWRVNHNiQ3aG43JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIrOjhEJSk+ISM1NyQkIitMTExMTCEjNSQiIiEhIiI3JCQiI0QhIiMkIitgel9WTiEjNTckJCIiIyEiIiQiIiEhIiI3JCQiK25tbW07ISM1JCIrJSpHJylcVyEjNTckJCIrSDlkRzkhIzUkIiIhISIiNyQkIiREIiEiJCQiK3dvSEhdISM1NyQkIis2NjY2NiEjNSQiIiEhIiI3JCQiIiIhIiIkIjEsKyt3TWlIYSEjOzckJCIrIjQ0NDQqISM2JCIiISEiIjckJCIrTExMTCQpISM2JCIqaENCcyYhIio3JCQiKyNwMkJwKCEjNiQiIiEhIiI3JCQiK1ZyJkc5KCEjNiQiKjRsYiVmISIqNyQkIitubW1tbSEjNiQiIiEhIiI3JCQiJEQnISIlJCIqbVg5NychIio3JCQiMjErKzUlSE4jKWUhIz0kIiIhISIiNyQkIitjYmJiYiEjNiQiK1BPaGppISM1NyQkIismKnk6al8hIzYkIiIhISIiNyQkIiImISIjJCIrMCpSNFEnISM1NyQkIitpWiE+dyUhIzYkIiIhISIiNyQkIitYWFhYWCEjNiQiK3VnVnprISM1NyQkIisoM0V5TSUhIzYkIiIhISIiNyQkIitubW1tVCEjNiQiKzMqPUxjJyEjNTckJCIiJSEiIyQiIiEhIiI3JCQiK1lROllRISM2JCIrKSlIak5tISM1NyQkIisvUHEuUCEjNiQiIiEhIiI3JCQiK3ImRzlkJCEjNiQiKzM4ailwJyEjNTckJCIraWVGW00hIzYkIiIhISIiNyQkIitMTExMTCEjNiQiMSoqKioqUnc7U3YnISM7NyQkIitfayFlQSQhIzYkIiIhISIiNyQkIiVESiEiJiQiKyRRK0ohbyEjNTckJCIqLi4uLiQhIzUkIiIhISIiNyQkIityazxUSCEjNiQiK056IXAlbyEjNTckJCIrZEc5ZEchIzYkIiIhISIiNyQkIit5eHh4RiEjNiQiK19ORCcpbyEjNTckJCIrLkZxLUYhIzYkIiIhISIiNyQkIitaKnk6aiMhIzYkIitaO3pAcCEjNTckJCIra0Q1a0QhIzYkIiIhISIiNyQkIiNEISIkJCIrI0hhUyZwISM1NyQkIipSQyFSQyEjNSQiIiEhIiI3JCQiKyJRXzRRIyEjNiQiK3ImeU0pcCEjNTckJCIrJlIiZURCISM2JCIiISEiIjckJCIrdHNzc0EhIzYkIiteb1U1cSEjNTckJCIrQUFBQUEhIzYkIiIhISIiNyQkIitWSSJSPCMhIzYkIio7LV8uKCEiKjckJCIrdSZmdzcjISM2JCIiISEiIjckJCIrTExMJDMjISM2JCIrTClmIWVxISM1NyQkIitGaiIzLyMhIzYkIiIhISIiNyQkIiIjISIjJCIrdGxAenEhIzU3JCQiKzlWeWc+ISM2JCIiISEiIjckJCIrQnAyQj4hIzYkIisxdiYpKTQoISM1NyQkIitgQ3onKT0hIzYkIiIhISIiNyQkIitfPSY9Jj0hIzYkIis2PDk8ciEjNTckJCIrPT09PT0hIzYkIiIhISIiNyQkIisnRzlkeSIhIzYkIismbzFVOCghIzU3JCQiK2xmUWE8ISM2JCIiISEiIjckJCIrSno4QzwhIzYkIit4PTxdciEjNTckJCIrYV8iXHAiISM2JCIiISEiIjckJCIrbm1tbTshIzYkIiteOjlsciEjNS0lJkNPTE9SRzYnJSRSR0JHJCIpQyllcSUhIikkIiIhISIiJCIoJz4hXCYhIiktJStfQVRUUklCVVRFRzYjLyUnc291cmNlRyUsbWF0aGRlZmF1bHRHLSUlVklFV0c2JDskIiIhISIiJCIiIiEiIjskIiImISIiJCIiKSEiIi0mJSZfQVhJU0c2IyIiIjYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEkYWJzNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSZmYWxzZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEkb3JkNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRTKCEiIi0lKUJPVU5EU19ZRzYjJCIkUyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVTSSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU/TSEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStTISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJStTISIiLSUpQ0hJTERSRU5HNiI=NiI=

n:=10:series(subs(P=Pser(n),HxP),x,2)

KydJInhHNiIsJComIyIjOyIiKiIiIikiIiQjRioiIiNGKkYqRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYu

evalf(-op(1,%)/GAMMA(-2/3))

JCIrTit1aXchIzU=

N:=60:i:=2: plot([seq([1/n,coeff(coeff(Cmser(N),x,-2*i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.1,ord=0.2..0.6,style=point);

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

n:=10:series(subs(P=Pser(n),HxP),x,3)

KylJInhHNiIsJComIyIjOyIiKiIiIikiIiQjRioiIiNGKkYqRiosJComIyIkNyIiIyIpRipGK0YqRipGLi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRiw=

evalf(-coeff(%,x,2)/GAMMA(-2/3))

JCIrOSozKmZmISM1

N:=80:i:=3: plot([seq([1/n,coeff(coeff(Cmser(N),x,-2*i),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.1,ord=0.2..0.6,style=point);

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Ig==

n:=10:series(subs(P=Pser(n),HxP),x,4)

KytJInhHNiIsJComIyIjOyIiKiIiIikiIiQjRioiIiNGKkYqRiosJComIyIkNyIiIyIpRipGK0YqRipGLiwkKiYjIiUzRSIlKD0jRipGK0YqRipGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqIiIl

evalf(-coeff(%,x,3)/GAMMA(-2/3))

JCIrbHMtU14hIzU=

Hxser:=proc(n) series(subs(P=Pser(n+2),HxP),x,n):end

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIeHNlckdGKGYqNiNJIm5HRihGKEYoRigtSSdzZXJpZXNHRiY2JS1JJXN1YnNHRiY2JC9JIlBHRigtSSVQc2VyR0YoNiMsJkYwIiIiIiIjRj1JJEh4UEdGKEkieEdGKEYwRihGKEYoNyNGLg==

Here we consider the series H(x,y)= sum H_{i,j} x^{i-j} y^j.In the paper we have replaced x by sqrt x HxP;HyL

LCQqLCMiI0siIioiIiIsJiomIiIkRidJIlBHNiJGJ0YnRichIiJGJylGKiNGJyIiI0YnLCZGK0YnRidGJ0YtKSwmRitGJ0YnRi1GMEYtRic=

LCQqMCMiIzsiIioiIiIpLCYqJiIiI0YnSSJMRzYiRidGJyIiJCEiIkYrRicpLCgqJClGLEYrRidGJyomIiM3RidGLEYnRi9GJkYnRitGJylGLiNGJ0YrRidGLEYnKSwmRixGJ0YuRi8iIiZGLywoRjJGJyomIiInRidGLEYnRidGJkYvRi9GLw==

Hfinal0:=(1/2*(1-(y+1/y)/4)*HyL-H00/y/8-1/4/x^2/y*HyL-HxP/4*(1/y+1/y/x^2))/(1-(y+x^2*y+1/y+1/x^2/y)/4)

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

Hfinal:=(x^2*H00/2+(1+x^2)*HxP+HyL*(1-2*x^2*y+x^2/2*(1+y^2)))/(1+x^2+x^4*y^2+x^2*y^2-4*x^2*y)

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

factor(Hfinal0-Hfinal)

IiIh

n:=6:ser:=map(Z->factor(expand(rationalize(Z))),mtaylor(subs(P=subs(x=x^2,Pser(n)),L=Lser(n),Hfinal),[x,y],n))

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

evalf(%)

LDokIit1R1NlaCEiKiIiIiomJCIrRipRNi8iISIpRiZJInlHNiJGJkYmKiYkIitQOT96SUYlRiYpSSJ4R0YsIiIjRiZGJiomJCIranpgKVEiRipGJilGK0YyRiZGJiooJCIrNjY2NnJGJUYmRjBGJkYrRiZGJiomJCIrTycpXCVwIkYqRiYpRisiIiRGJkYmKiYkIitdVyRcUiNGJUYmKUYxIiIlRiZGJiooJCIrSCgpMyEzIkYqRiZGMEYmRjZGJkYmKiYkIit5cCFSKD5GKkYmKUYrRkNGJkYmKigkIitOWXAwX0YlRiZGK0YmRkJGJkYmKigkIitVckQ0OUYqRiZGPUYmRjBGJkYmKiYkIitwSkRNQUYqRiYpRisiIiZGJkYm

Non-boundary coefficients serdiff:=mtaylor(ser-subs(x=x^2,Hxser(n)+Hyser(n)),[x,y],10)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzZXJkaWZmR0YoLCoqKCMiI2siIioiIiIpSSJ4R0YoIiIjRjNJInlHRihGM0YzKiojIiNLIiMiKUYzLCYqJiIiJkYzKSIiJCNGM0Y2RjNGMyIjTyEiIkYzRjRGMylGN0Y2RjNGQyoqI0YxIiRWI0YzLCYiJCo+RkMqJiIjJSlGM0Y/RjNGM0YzKUY3RkBGM0Y0RjNGQyoqRjBGMywmKiRGP0YzRjNGM0ZDRjNGN0YzKUY1IiIlRjNGMzcjRi4=

let us check these coefficients N:=80:k:=2:l:=1: plot([seq([1/n,coeff(coeff(coeff(convert(Cser(N),polynom),x,l-k),y,l),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=1.4..1.8,style=point);

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

evalf(-coeff(coeff(serdiff,x,k),y,l)/GAMMA(-2/3))

JCIrNFNqcDwhIio=

k:=2:l:=2: evalf(-coeff(coeff(serdiff,x,k),y,l)/GAMMA(-2/3))

JCIrZUMmeW8jISIq

N:=80: plot([seq([1/n,coeff(coeff(coeff(convert(Cser(N),polynom),x,l-k),y,l),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=2...2.7,style=point);

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NiI=

serdiff

LCoqKCMiI2siIioiIiIpSSJ4RzYiIiIjRidJInlHRipGJ0YnKiojIiNLIiMiKUYnLCYqJiIiJkYnKSIiJCNGJ0YrRidGJyIjTyEiIkYnRihGJylGLEYrRidGOCoqI0YlIiRWI0YnLCYiJCo+RjgqJiIjJSlGJ0Y0RidGJ0YnKUYsRjVGJ0YoRidGOCoqRiRGJywmKiRGNEYnRidGJ0Y4RidGLEYnKUYpIiIlRidGJw==

k:=4:l:=1: evalf(-coeff(coeff(serdiff,x,k),y,l)/GAMMA(-2/3))

JCIrMz9ZJkgiISIq

N:=80: plot([seq([1/n,coeff(coeff(coeff(convert(Cser(N),polynom),x,l-k),y,l),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=1...1.3,style=point);

NistJSdDVVJWRVNHNiQ3XHA3JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIiISEiIjckJCIrTExMTEwhIzUkIislZXkrJj4hIzU3JCQiI0QhIiMkIiIhISIiNyQkIiIjISIiJCIrKT4xXCZRISM1NyQkIitubW1tOyEjNSQiIiEhIiI3JCQiK0g5ZEc5ISM1JCIrZkxQJUcmISM1NyQkIiREIiEiJCQiIiEhIiI3JCQiKzY2NjY2ISM1JCIxKioqKipcdCpRXWohIzs3JCQiIiIhIiIkIiIhISIiNyQkIisiNDQ0NCohIzYkIjEsKys6L3ZrciEjOzckJCIrTExMTCQpISM2JCIiISEiIjckJCIrI3AyQnAoISM2JCIqclZNIXkhIio3JCQiK1ZyJkc5KCEjNiQiIiEhIiI3JCQiK25tbW1tISM2JCIrMjZIOyQpISM1NyQkIiREJyEiJSQiIiEhIiI3JCQiMjErKzUlSE4jKWUhIz0kIitgNFtPKCkhIzU3JCQiK2NiYmJiISM2JCIiISEiIjckJCIrJip5OmpfISM2JCIrblVyJzMqISM1NyQkIiImISIjJCIiISEiIjckJCIraVohPnclISM2JCIrO20lSFEqISM1NyQkIitYWFhYWCEjNiQiIiEhIiI3JCQiKygzRXlNJSEjNiQiKkR6bWoqISIqNyQkIitubW1tVCEjNiQiIiEhIiI3JCQiIiUhIiMkIit6SVJjKSohIzU3JCQiK1lROllRISM2JCIiISEiIjckJCIrL1BxLlAhIzYkIiticCVbKyIhIio3JCQiK3ImRzlkJCEjNiQiIiEhIiI3JCQiK2llRltNISM2JCIrVyh6PC0iISIqNyQkIitMTExMTCEjNiQiIiEhIiI3JCQiK19rIWVBJCEjNiQiK1IjPW8uIiEiKjckJCIlREohIiYkIiIhISIiNyQkIiouLi4uJCEjNSQiK2FIRV01ISIqNyQkIityazxUSCEjNiQiIiEhIiI3JCQiK2RHOWRHISM2JCIrX1VOaTUhIio3JCQiK3l4eHhGISM2JCIiISEiIjckJCIrLkZxLUYhIzYkIis+bUd0NSEiKjckJCIrWip5OmojISM2JCIiISEiIjckJCIra0Q1a0QhIzYkIit2IT5LMyIhIio3JCQiI0QhIiQkIiIhISIiNyQkIipSQyFSQyEjNSQiKilHRyM0IiEiKTckJCIrIlFfNFEjISM2JCIiISEiIjckJCIrJlIiZURCISM2JCIrQHRlKzYhIio3JCQiK3Rzc3NBISM2JCIiISEiIjckJCIrQUFBQUEhIzYkIitAVEEzNiEiKjckJCIrVkkiUjwjISM2JCIiISEiIjckJCIrdSZmdzcjISM2JCIrPDNGOjYhIio3JCQiK0xMTCQzIyEjNiQiIiEhIiI3JCQiK0ZqIjMvIyEjNiQiKyhSJHpANiEiKjckJCIiIyEiIyQiIiEhIiI3JCQiKzlWeWc+ISM2JCIrdiRbeTciISIqNyQkIitCcDJCPiEjNiQiIiEhIiI3JCQiK2BDeicpPSEjNiQiK0JXW0w2ISIqNyQkIitfPSY9Jj0hIzYkIiIhISIiNyQkIis9PT09PSEjNiQiKyhvVihRNiEiKjckJCIrJ0c5ZHkiISM2JCIiISEiIjckJCIrbGZRYTwhIzYkIitYR21WNiEiKjckJCIrSno4QzwhIzYkIiIhISIiNyQkIithXyJccCIhIzYkIitqUkZbNiEiKjckJCIrbm1tbTshIzYkIiIhISIiNyQkIitpVU1SOyEjNiQiKV9nXzYhIig3JCQiK0VLIUhoIiEjNiQiIiEhIiI3JCQiKyhlLHRlIiEjNiQiKyNRIm9jNiEiKjckJCImRGMiISInJCIiISEiIjckJCIrUTpZUTohIzYkIisjW0MwOyIhIio3JCQiKzo6Ojo6ISM2JCIiISEiIjckJCIrOHRgI1wiISM2JCIrPlM6azYhIio3JCQiK04jKWVxOSEjNiQiIiEhIiI3JCQiK2lgRlw5ISM2JCIrKlEoZW42ISIqNyQkIitIOWRHOSEjNiQiIiEhIiI3JCQiKy8yWDM5ISM2JCIrXCwlMzwiISIqNyQkIisqKSkpKSkpUSIhIzYkIiIhISIiNyQkIis5SScpcDghIzYkIiteaSNSPCIhIio3JCQiK144Tl44ISM2JCIiISEiIjckJCIrTExMTDghIzYkIitXI2VvPCIhIio3JCQiK3UlKnk6OCEjNiQiIiEhIiI3JCQiKypILCgpSCIhIzYkIitcdWt6NiEiKjckJCIrI0deP0ciISM2JCIiISEiIjckJCIrJnlBZUUiISM2JCIpVEkjPSIhIig3JCQiJEQiISIlJCIiISEiIi0lJkNPTE9SRzYnJSRSR0JHJCIpQyllcSUhIikkIiIhISIiJCIoJz4hXCYhIiktJStfQVRUUklCVVRFRzYjLyUnc291cmNlRyUsbWF0aGRlZmF1bHRHLSUlVklFV0c2JDskIiIhISIiJCIiJiEiIzskIiM1ISIiJCIjOCEiIi0mJSZfQVhJU0c2IyIiIjYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEkYWJzNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSZmYWxzZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEkb3JkNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRTKCEiIi0lKUJPVU5EU19ZRzYjJCIkUyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVTSSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU/TSEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStTISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJStTISIiLSUpQ0hJTERSRU5HNiI=NiI=

serdiff

LCoqKCMiI2siIioiIiIpSSJ4RzYiIiIjRidJInlHRipGJ0YnKiojIiNLIiMiKUYnLCYqJiIiJkYnKSIiJCNGJ0YrRidGJyIjTyEiIkYnRihGJylGLEYrRidGOCoqI0YlIiRWI0YnLCYiJCo+RjgqJiIjJSlGJ0Y0RidGJ0YnKUYsRjVGJ0YoRidGOCoqRiRGJywmKiRGNEYnRidGJ0Y4RidGLEYnKUYpIiIlRidGJw==

k:=2:l:=3: evalf(-coeff(coeff(serdiff,x,k),y,l)/GAMMA(-2/3))

JCIrJ3oucV0kISIq

N:=100: plot([seq([1/n,coeff(coeff(coeff(convert(Cser(N),polynom),x,l-k),y,l),t,n)/4^(n)*n^(5/3)],n=1..N)],abs=0..0.05,ord=2.5...3.5,style=point);

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T1VORFNfV0lEVEhHNiMkIiUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNISUxEUkVORzYiRzYi

JSFH JSFH