This Maple session accompanies the paper "Three quadrant walks by invariants" and considers the case of Kreweras' steps.The results are stated in the introduction of the paper, and the proofs are in Section 4.

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

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

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Cmser:=proc(n) local i, res: res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: C00ser:=proc(n) coeff(coeff(convert(Cser(n),polynom),y,0),x,0): end: Cser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C(x,y)=Cser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),eqC+C(x,y)),t,n+1)): fi: end: Cmser(8); C00ser(8)

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LCgqJiIjWSIiIilJInRHNiIiIidGJUYlKiYiIiVGJSlGJyIiJEYlRiVGJUYl

map(expand,Cser(4));

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We separate in C the contribution of two regions U denotes the part of C above the diagonal, DD the diagonal itself User:=proc(n) local i,j: collect(add(add(coeff(coeff(convert(Cser(n),polynom),x,i),y,j)*x^(j-i-1)*y^(j),i=-n..j-1),j=0..n+4),t,normal): end: User(5);

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

LCoiIiJGIyomLCYqJClJInlHNiIiIiRGI0YjIiIlRiNGIylJInRHRilGKkYjRiMqJilGKCIiI0YjKUYtRjBGI0YjKiZGKEYjRi1GI0Yj

Cexpr:=DD(x*y)+1/x*U(1/x,x*y)+1/y*U(1/y,x*y);

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The original model, and the companion model Steps:=x*y+1/x+1/y; tSteps:=expand(subs({y=y*x,x=1/x},Steps));

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The polynomials V for the companion model Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWMHlHRihJInlHRig3I0Yu

The polynomial Hminus for the original model Hmx:=coeff(Steps, y,-1);

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Generic equation relating D and U eqD:=-(1-t*V0y)*DD(y)+1+2*t*Vmy*U(0,y)-2*t/y*U(0,0);

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First equation for U eqU:=-x*(1-t*tSteps)*U(x,y)+t*x*Vpy*DD(y)-t/y*subs(x=1/x,Hmx)*U(x,0)-t*Vmy*U(0,y)+t/y*U(0,0);

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Elimination of U(0,y): the equation of Lemma 2.1 eqUfinal:=collect(subs(isolate(eqD,U(0,y)),eqU*y*2),[U,DD],factor);

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Let us check n:=10: normal(series(subs( U(x,y)=User(n), U(x,0)=subs(y=0,User(n)), DD(y)=Dser(n),eqUfinal),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Kr:=1-t*tSteps

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

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Qser:=proc(n) option remember: if n=0 then 1 else 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)): fi: end: map(expand,Qser(4));

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Two pairs of invariants (Table 3). The first one is rational I0x:=1/x+x/t-x^2; J0y:=subs(x=y,I0x);

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Let us check the ratio ratio0:=factor((I0x-J0y)/Kr);

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Decoupling for xy fx:=x/t-x^2; gy:=-1/y

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

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LCQqJkkieEc2IiIiIkkidEdGJSEiIkYo

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

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

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ratio1:=factor(%/Kr);

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

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0U3RlcHNHRigsKEkieEdGKCIiIiomSSJ5R0YoISIiRi9GM0YwRjJGMDcjRi4=

Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

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The tricky coefficient in the functional equation satisfied by U(x,y) co:=t*V0y+2*t*x*Vpy-1;

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The functions F and G and H of Table 5 Fx:=-2*x+1/t; Gy:=1/t;

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Hxy:=factor(y-Gy*co-Fx);

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We can thus construct new invariants, as in Section 3.2 Sy:=y*DD(y)+Gy;

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Rx:=2*t*subs(x=1/x,Hmx)*U(x,0)-Fx;

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Delta:=(1-t*V0y)^2-4*t^2*Vpy*Vmy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwmKiQpLCYqJkkidEdGKCIiIkkieUdGKEY0ISIiRjRGNCIiI0Y0RjQqKCIiJUY0KUYzRjdGNEY1RjZGNjcjRi4=

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

LCQqKEkidEc2IiEiIixCKigpLUkjRERHRiU2I0kieUdGJSIiIyIiIilGJCIiJEYvKUYtIiImRi9GLyoqRi5GL0YpRi8pRiRGLkYvKUYtIiIlRi9GJioqRjdGL0YpRi9GMEYvKUYtRi5GL0YmKipGLkYvRipGL0Y1Ri9GNkYvRi8qKEYpRi9GJEYvKUYtRjFGL0YvKipGN0YvKS1JIlVHRiU2JEkieEdGJSIiIUYuRi9GMEYvRi1GL0YmKipGN0YvRipGL0YkRi9GPEYvRiYqLCIiKUYvRj9GL0Y1Ri9GQkYvRi1GL0YmKipGRkYvRipGL0Y1Ri9GLUYvRiYqKkY3Ri8pRkJGLkYvRi1GL0YkRi9GJiomRiRGL0Y8Ri9GLyooRi5GL0YqRi9GOUYvRi8qKkY3Ri9GP0YvRiRGL0YtRi9GLyooRjdGL0ZCRi9GLUYvRi8qJkYuRi9GOUYvRiYqJkY3Ri9GJEYvRiZGL0YtRiZGJg==

Let us check that this is divisible by Kr(x,y). For this, we use the functional equation satisfied by U(x,y) to express U(x,0) factor(subs(isolate(eqUfinal,U(x,0)),%)/Kr);

LCQqKiIiJSIiIkkieEc2IkYlLDwqKiktSSJVR0YnNiRGJkkieUdGJyIiI0YlKUkidEdGJ0YvRiUpRiZGL0YlKUYuRi9GJUYlKipGKkYlRjBGJUYmRiUpRi4iIiRGJUYlKi5GL0YlRitGJS1JI0RER0YnNiNGLkYlRjBGJUYmRiVGM0YlRiUqKkYrRiVGOEYlRjBGJUY1RiVGJSoqRipGJUYxRiVGJkYlRjNGJSEiIiooKUY4Ri9GJUYwRiVGM0YlRiUqKEYqRiVGMEYlRi5GJUYlKipGK0YlRjhGJUYxRiVGM0YlRj0qLEYvRiVGK0YlRjFGJUYmRiVGLkYlRiUqKEYrRiVGMUYlRjNGJUYlKipGL0YlRi5GJUY4RiVGMUYlRiUqJkYrRiVGLkYlRj1GJUYlRiVGMUY9Rj0=

Let us check the expression of the ratio given in the proposition: this should be 0 factor(%- (-4*t*x*Vpy*Sy^2+(Hxy-2*x*y*U(x,y))*((t*V0y+2*t*x*Vpy-1)*Sy+Rx)))

LCQqLCIiIyIiIkkieEc2IkYlLUkiVUdGJzYkRiZJInlHRidGJUYrRiUsNCosRiRGJUYoRiVJInRHRidGJSlGJkYkRiVGK0YlRiUqLEYkRiVGKEYlRi5GJUYmRiUpRitGJEYlRiUqLEYkRiUtSSNEREdGJzYjRitGJUYuRiVGJkYlRitGJUYlKihGM0YlRi5GJUYxRiVGJSoqRiRGJUYmRiVGK0YlRihGJSEiIiooRiRGJUYoRiVGLkYlRiUqJkYrRiVGM0YlRjgqKEYkRiVGLkYlLUYpNiRGJiIiIUYlRjhGK0YlRiVGOA==

factor(subs(isolate(eqUfinal,U(x,0)),%));

IiIh

The new invariants Ix:=Rx^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCokKSwoKigiIiMiIiJJInRHRihGMy1JIlVHRig2JEkieEdGKCIiIUYzRjMqJkYyRjNGOEYzRjMqJEY0ISIiRjxGMkYzNyNGLg==

Jy:=Delta*Sy^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKComLCYqJCksJiomSSJ0R0YoIiIiSSJ5R0YoRjUhIiJGNUY1IiIjRjVGNSooIiIlRjUpRjRGOEY1RjZGN0Y3RjUpLCYqJkY2RjUtSSNEREdGKDYjRjZGNUY1KiRGNEY3RjVGOEY1NyNGLg==

The ratio (Ix-Jy)/Kr, given in Proposition 3.2 ratio :=-4*t*x*Vpy*Sy^2+(Hxy-2*x*y*U(x,y))*((t*V0y+2*t*x*Vpy-1)*Sy+Rx);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyYXRpb0dGKCwmKioiIiUiIiJJInRHRihGMUkieEdGKEYxKSwmKiZJInlHRihGMS1JI0RER0YoNiNGN0YxRjEqJEYyISIiRjEiIiNGMUY8KixGPUYxRjNGMUY3RjEtSSJVR0YoNiRGM0Y3RjEsKiomLCgqKEY9RjFGMkYxRjNGMUYxKiZGMkYxRjdGMUYxRjFGPEYxRjVGMUYxKihGPUYxRjJGMS1GQDYkRjMiIiFGMUYxKiZGPUYxRjNGMUYxRjtGPEYxRjw3I0Yu

It is a multiple of x series(ratio,x,1);

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

But not of y series(ratio,y,1);

KydJInlHNiIsJCooIiIlIiIiSSJ4R0YkRihJInRHRiQhIiJGKyIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

I0x; J0y;

LCgqJEkieEc2IiEiIiIiIiomRiRGJ0kidEdGJUYmRicqJClGJCIiI0YnRiY=

LCgqJEkieUc2IiEiIiIiIiomRiRGJ0kidEdGJUYmRicqJClGJCIiI0YnRiY=

I1x;J1y;ratio1;

LCgqJkkidEc2IiIiIi1JIlFHRiU2JEkieEdGJSIiIUYmRiYqJkYqRiZGJCEiIkYtKiQpRioiIiNGJkYm

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKiZGJEYmLUYoNiRGKkYqRiZGJiokRitGLEYs

LCQqKEkieEc2IiIiIiwmKigtSSJRR0YlNiRGJEkieUdGJUYmSSJ0R0YlRiZGLEYmRiZGJkYmRiZGLSEiIkYu

series(ratio1,x,1);series(ratio1,y,1);

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

KydJInlHNiIsJComSSJ4R0YkIiIiSSJ0R0YkISIiRioiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

series(ratio-4*ratio1,x,1); series(ratio-4*ratio1,y,1);

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

The following two series should be independent of x and y (and equal) Ix-4*I1x; Jy-4*J1y;

LCoqJCksKCooIiIjIiIiSSJ0RzYiRigtSSJVR0YqNiRJInhHRioiIiFGKEYoKiZGJ0YoRi5GKEYoKiRGKSEiIkYyRidGKEYoKigiIiVGKEYpRigtSSJRR0YqRi1GKEYyKihGNEYoRi5GKEYpRjJGKComRjRGKClGLkYnRihGMg==

LCoqJiwmKiQpLCYqJkkidEc2IiIiIkkieUdGKkYrISIiRitGKyIiI0YrRisqKCIiJUYrKUYpRi5GK0YsRi1GLUYrKSwmKiZGLEYrLUkjRERHRio2I0YsRitGKyokRilGLUYrRi5GK0YrKihGMEYrRilGKy1JIlFHRio2JCIiIUYsRitGKyooRjBGK0YpRistRjs2JEY9Rj1GK0YtKiZGMEYrRixGLUYr

Let us check this. This gives the first terms of the unknown series A n:=20: Aexpser:=normal(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)), Ix-4*I1x),t,n));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShBZXhwc2VyR0YoKzNJInRHRigiIiIhIiMhIilGMCEjSyIiJSEkbyQiIighJVtjIiM1ISZ3KCoqIiM4IShjcSI+IiM7LUkiT0dGJTYjRjAiIz43I0Yu

n:=20: normal(series(subs(DD(y)=Dser(n), Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)), Jy-4*J1y),t,n));

KzNJInRHNiIiIiIhIiMhIilGJSEjSyIiJSEkbyQiIighJVtjIiM1ISZ3KCoqIiM4IShjcSI+IiM7LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIz4=

The solution of quadrant walks with reverse Kreweras steps, taken from earlier papers algV:=-V+t*(2+V^3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRighIiIqJkkidEdGKCIiIiwmKiQpRi8iIiRGM0YzIiIjRjNGM0YzNyNGLg==

tV:=solve(algV,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0VkdGKComSSJWR0YoIiIiLCYqJClGLyIiJEYwRjAiIiNGMCEiIjcjRi4=

with(gfun): Vser:=proc(n) op(algeqtoseries(algV,t,V,n,true)): end: Vser(4); Vser(5);

KylJInRHNiIiIiMiIiIiIikiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJg==

KylJInRHNiIiIiMiIiIiIikiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

Deltapx:=1-V*(4+V^3)/4*x+V^2/4*x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEZWx0YXB4R0YoLCgiIiJGLyoqI0YvIiIlRi9JIlZHRihGLywmKiQpRjMiIiRGL0YvRjJGL0YvSSJ4R0YoRi8hIiIqKEYxRi8pRjMiIiNGLylGOEY8Ri9GLzcjRi4=

Qx0:=V*(4-V^3)/16/t-(t-x^2+t*x^3)/2/x/t^2- (2*x-V^2-V/x)*sqrt(Deltapx)/2/t/V;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRReDBHRigsKCoqIyIiIiIjO0YxSSJWR0YoRjEsJiokKUYzIiIkRjEhIiIiIiVGMUYxSSJ0R0YoRjhGMSoqI0YxIiIjRjEsKComRjpGMSlJInhHRihGN0YxRjEqJClGQUY9RjFGOEY6RjFGMUZBRjgpRjpGPUY4RjgqLCNGMUY5RjEsKComRj1GMUZBRjFGMSokKUYzRj1GMUY4KiZGM0YxRkFGOEY4RjEpLCoqJilGM0Y5RjFGQUYxRjgqJkZKRjFGQ0YxRjEqKEY5RjFGM0YxRkFGMUY4RjlGMUY8RjFGOkY4RjNGOEY4NyNGLg==

Let us check n:=12: factor(series(subs(V=Vser(n),Qx0),t,n));

KzlJInRHNiIiIiIiIiFJInhHRiRGJSokKUYnIiIjRiVGKiwmKiQpRiciIiRGJUYlRipGJUYuKiZGJ0YlLCZGLEYlIiImRiVGJSIiJSomRilGJSwmRixGJSIiKkYlRiVGMSwoKiQpRiciIidGJUYlKiYiIzlGJUYtRiVGJSIjO0YlRjkqKEYnRiUsJkYsRiVGLkYlRiUsJkYsRiUiIzxGJUYlIiIoKiZGKUYlLChGN0YlKiYiI0ZGJUYtRiVGJSIkOiJGJUYlIiIpLCoqJClGJ0Y1RiVGJSomIiNORiVGOEYlRiUqJiIkPyNGJUYtRiVGJSIkIz5GJUY1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIzU=

map(factor,subs(y=0,Qser(9)));

KzdJInRHNiIiIiIiIiFJInhHRiRGJSokKUYnIiIjRiVGKiwmKiQpRiciIiRGJUYlRipGJUYuKiZGJ0YlLCZGLEYlIiImRiVGJSIiJSomRilGJSwmRixGJSIiKkYlRiVGMSwoKiQpRiciIidGJUYlKiYiIzlGJUYtRiVGJSIjO0YlRjkqKEYnRiUsJkYsRiVGLkYlRiUsJkYsRiUiIzxGJUYlIiIoKiZGKUYlLChGN0YlKiYiI0ZGJUYtRiVGJSIkOiJGJUYlIiIpLCoqJClGJ0Y1RiVGJSomIiNORiVGOEYlRiUqJiIkPyNGJUYtRiVGJSIkIz5GJUY1

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

factor(series(Qx0,x,2));

KydJInhHNiIsJCoqIyIiIiIiKUYoSSJWR0YkRigsJiokKUYqIiIkRihGKCIiJSEiIkYoSSJ0R0YkRjBGMCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

Q00:=convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRMDBHRigsJCoqIyIiIiIiKUYxSSJWR0YoRjEsJiokKUYzIiIkRjFGMSIiJSEiIkYxSSJ0R0YoRjlGOTcjRi4=

The expressions of Ix and Jy Ixexpr:=4*I1x+A; Jyexpr:=4*J1y+A;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJeGV4cHJHRigsKiooIiIlIiIiSSJ0R0YoRjEtSSJRR0YoNiRJInhHRigiIiFGMUYxKihGMEYxRjZGMUYyISIiRjkqJkYwRjEpRjYiIiNGMUYxSSJBR0YoRjE3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeWV4cHJHRigsKiooIiIlIiIiSSJ0R0YoRjEtSSJRR0YoNiQiIiFJInlHRihGMSEiIiooRjBGMUYyRjEtRjQ2JEY2RjZGMUYxKiZGMEYxRjdGOEY4SSJBR0YoRjE3I0Yu

To determine the value of this constant A, we cancel Delta by specializing y to V^2 n:=10: normal(series(subs(y=Vser(n)^2, Delta),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

J(y) thus vanishes at V^2 subs(Q(0,y)=subs(x=y,Qx0),Q(0,0)=Q00,y=V^2,t=tV,Jyexpr);

LCoqKiIiJSIiIkkiVkc2IkYlLCYqJClGJiIiJEYlRiUiIiNGJSEiIiwoKigjRiUiIztGJSwmRilGLUYkRiVGJUYoRiVGJSoqI0YlRixGJSwoKiYpRiYiIihGJUYoRi1GJSokKUYmRiRGJUYtKiZGJkYlRihGLUYlRiVGOkYtKUYoRixGJUYtKiwjRiVGJEYlLCYqJClGJkYsRiVGJSokRiZGLUYtRiUpLCYqJkYkRiVGKkYlRi1GJEYlRjRGJUZBRi1GKEYlRi1GJUYtKihGNEYlRiZGJSwmRilGJUYkRi1GJUYtKiZGJEYlRkFGLUYtSSJBR0YnRiU=

Asol:=simplify(solve(%,A));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBc29sR0YoLCQqKCMiIiIiIiVGMSwsKiQpSSJWR0YoIiInRjFGMSooIiIpRjEpLCYqJClGNiIiJEYxISIiRjFGMSNGMSIiI0YxRj1GMUY/KiYiIzdGMUY9RjFGMSomRjlGMUY6RjFGMUY5RjFGMSlGNkZBRj9GMTcjRi4=

Asol:=subs(delta=1-V^3,collect(subs(sqrt(1-V^3)=sqrt(delta),Asol),delta,factor))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBc29sR0YoLCYqLCIiIyIiIiwmSSJWR0YoRjFGMSEiIkYxLCgqJClGM0YwRjFGMUYzRjFGMUYxRjFGN0Y0KSwmKiQpRjMiIiRGMUY0RjFGMSNGMUYwRjFGNCooI0YxIiIlRjEsKCokKUYzIiInRjFGMSomIiM3RjFGO0YxRjEiIilGMUYxRjdGNEYxNyNGLg==

Let us check n:=20:normal(series(subs(V=Vser(n),Asol),t,n));

KzNJInRHNiIiIiIhIiMhIilGJSEjSyIiJSEkbyQiIighJVtjIiM1ISZ3KCoqIiM4IShjcSI+IiM7LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIz0=

Aexpser;

KzNJInRHNiIiIiIhIiMhIilGJSEjSyIiJSEkbyQiIighJVtjIiM1ISZ3KCoqIiM4IShjcSI+IiM7LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIz4=

Let us get back to the expressions of invariants. First Ix Ixexpr;

LCoqKCIiJSIiIkkidEc2IkYlLUkiUUdGJzYkSSJ4R0YnIiIhRiVGJSooRiRGJUYrRiVGJiEiIkYuKiZGJEYlKUYrIiIjRiVGJUkiQUdGJ0Yl

expand(subs(Q(x,0)=Qx0,A=Asol,t=tV,Ixexpr));

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

Ixfinal:=collect(subs(sqrt(-V^4*x+V^2*x^2-4*V*x+4)=2*sqrt(Dpx),sqrt(-V^3+1)=sqrt(delta),%),[delta,Dpx],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShJeGZpbmFsR0YoLCgqLCIiIyIiIiwmSSJWR0YoRjFGMSEiIkYxLCgqJClGM0YwRjFGMUYzRjFGMUYxRjFGN0Y0KUkmZGVsdGFHRigjRjFGMEYxRjQqLEYwRjEsKComRjdGMUkieEdGKEYxRjEqJkYwRjEpRj5GMEYxRjRGM0YxRjFGM0Y0Rj5GNClJJERweEdGKEY6RjFGMSosRjBGMSwmRjZGMUY+RjRGMSksJiomRjNGMUY+RjFGMUYxRjRGMEYxRj5GNEY3RjRGNDcjRi4=

Expression of the paper for Ix/2 exprpap:=(1-V^3)^(3/2)/V^2+(1-x*V)^2*(1/V^2-1/x)+(1/x+V-2*x/V)* sqrt(Dpx)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHBycGFwR0YoLCgqJiksJiokKUkiVkdGKCIiJCIiIiEiIkY2RjYjRjUiIiNGNilGNEY5RjdGNiomKSwmKiZGNEY2SSJ4R0YoRjZGN0Y2RjZGOUY2LCYqJEY6RjdGNiokRj9GN0Y3RjZGNiomLChGQkY2RjRGNiooRjlGNkY/RjZGNEY3RjdGNilJJERweEdGKCNGNkY5RjZGNjcjRi4=

factor(subs(delta=1-V^3,exprpap-Ixfinal/2))

IiIh

collect((-V^4*x+V^2*x^2-4*V*x+4)/4,x,factor)

LCgiIiJGIyoqI0YjIiIlRiNJIlZHNiJGIywmKiQpRiciIiRGI0YjRiZGI0YjSSJ4R0YoRiMhIiIqKEYlRiMpRiciIiNGIylGLUYxRiNGIw==

subs(U(x,0)=1/x*Cm(x),Ix/2)

LCQqJiMiIiIiIiNGJSksKCoqRiZGJUkidEc2IkYlSSJ4R0YrISIiLUkjQ21HRis2I0YsRiVGJSomRiZGJUYsRiVGJSokRipGLUYtRiZGJUYl

In the first theorem of the paper we give I(x) /2 n:=10: normal(series(subs(delta=1-V^3,Dpx=Deltapx,V=Vser(n),Ixfinal),t,n));

KzlJInRHNiIiIiIhIiMsJComIiIlRiVJInhHRiRGJSEiIkYrLCQqJkYpRiUpRioiIiNGJUYlIiIhISIlRiUsJEYoRiVGL0YsIiIkLCYqJkYpRiUpRipGM0YlRiUiI0NGK0YpLCYqJkYpRiUpRipGKUYlRiUqJiIjP0YlRipGJUYlIiImLCQqKEYpRiVGLkYlLCYqJEY2RiVGJSIiKkYlRiVGJSIiJywoKiZGKUYlKUYqRkNGJUYlKiYiI2NGJUY2RiVGJSIkLyRGKyIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIp

normal(series(subs(U(x,0)=subs(y=0,User(n)),Ix),t,n));

KztJInRHNiIiIiIhIiMsJComIiIlRiVJInhHRiRGJSEiIkYrLCQqJkYpRiUpRioiIiNGJUYlIiIhISIlRiUsJEYoRiVGL0YsIiIkLCYqJkYpRiUpRipGM0YlRiUiI0NGK0YpLCYqJkYpRiUpRipGKUYlRiUqJiIjP0YlRipGJUYlIiImLCYqJkYpRiUpRipGPUYlRiUqJiIjT0YlRi5GJUYlIiInLCgqJkYpRiUpRipGQ0YlRiUqJiIjY0YlRjZGJUYlIiQvJEYrIiIoLCgqJkYpRiUpRipGSkYlRiUqJiIjISlGJUY6RiVGJSomIiQvI0YlRipGJUYlIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIio=

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

Now J(y) expand(subs(Q(0,y)=subs(x=y,Qx0),Q(0,0)=Q00,A=Asol,Jyexpr));

LDYqJiIiJSIiIkkiVkc2IkYlRiUqJiIiI0YlKUkieUdGJ0YpRiVGJSooRilGJUYrRiVJInRHRichIiJGLiomRilGJUYrRi5GLioqRilGJSksKiomKUYmRiRGJUYrRiVGLiomKUYmRilGJUYqRiVGJSooRiRGJUYmRiVGK0YlRi5GJEYlI0YlRilGJUYmRi5GK0YlRiUqJkYxRiVGJkYlRi4qJkYxRiVGK0YuRi4qKEYpRiVGJkYlKSwmKiQpRiYiIiRGJUYuRiVGJUY4RiVGLiooRilGJUY2Ri5GPEYlRiUqJkYpRiVGNkYuRiU=

collect(subs(sqrt(-V^4*y+V^2*y^2-4*V*y+4)=2*sqrt(Dpy),sqrt(-V^3+1)=sqrt(delta),%),[delta,Dpy],factor);

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieUdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweUdGKEYvRiVGKSosRiRGJSwsKihGLEYlSSJ0R0YoRiUpRjMiIiRGJUYlKipGJEYlKUYnRj1GJUY7RiVGM0YlRiUqJkYsRiVGNUYlRikqJkYsRiVGO0YlRikqJkY7RiVGM0YlRiVGJUY7RilGM0YpRixGKUYl

Jyfinal:=subsop(3=factor(subs(t=tV,op(3,%))),%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShKeWZpbmFsR0YoLCgqLCIiIyIiIiwmSSJWR0YoRjFGMSEiIkYxLCgqJClGM0YwRjFGMUYzRjFGMUYxRjFGN0Y0KUkmZGVsdGFHRigjRjFGMEYxRjQqLEYwRjEsKComRjdGMUkieUdGKEYxRjEqJkYwRjEpRj5GMEYxRjRGM0YxRjFGM0Y0Rj5GNClJJERweUdGKEY6RjFGNCosRjBGMSwmRjZGMUY+RjRGMSksJiomRjNGMUY+RjFGMUYxRjRGMEYxRjdGNEY+RjRGNDcjRi4=

In the paper we give J(x)/2 n:=12: factor(series(subs(delta=1-V^3, Dpy=subs(x=y,Deltapx),V=Vser(n),Jyfinal),t,n));

KztJInRHNiIiIiIhIiMsJComIiIlRiVJInlHRiQhIiJGKyIiISEiKUYlLCQqJkYpRiVGKkYlRisiIiMsJComRilGJSlGKkYwRiVGKyIiJCwkKihGKUYlLCZGKkYlRjBGJUYlLCgqJEYzRiVGJSomRjBGJUYqRiVGK0YpRiVGJUYrRiksJCooRilGJUYqRiUsJiokKUYqRjRGJUYlIiImRiVGJUYrRkAsJCooRilGJUYzRiUsJkY+RiUiIipGJUYlRisiIicsKComRilGJSlGKkZFRiVGKyomIiNjRiVGP0YlRisiJG8kRisiIigsJCoqRilGJUYqRiUsJkY+RiVGNEYlRiUsJkY+RiUiIzxGJUYlRisiIiksJCooRilGJUYzRiUsKCokRkhGJUYlKiYiI0ZGJUY/RiVGJSIkOiJGJUYlRitGRC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM1

factor(series(subs(DD(y)=Dser(n),Jy),t,n));

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

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Jy/2

LCQqKCMiIiIiIiNGJSwmKiQpLCYqJkkidEc2IkYlSSJ5R0YtRiUhIiJGJUYlRiZGJUYlKigiIiVGJSlGLEYmRiVGLkYvRi9GJSksJiomRi5GJS1JI0RER0YtNiNGLkYlRiUqJEYsRi9GJUYmRiVGJQ==

Just a sign between the two values Ixfinal; Jyfinal;

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieEdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweEdGKEYvRiVGJSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRjNGKUYsRilGKQ==

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieUdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweUdGKEYvRiVGKSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRixGKUYzRilGKQ==

Ixfinal-subs(y=x,Dpy=Dpx,Jyfinal);

LCQqLCIiJSIiIiwoKiYpSSJWRzYiIiIjRiVJInhHRipGJUYlKiZGK0YlKUYsRitGJSEiIkYpRiVGJUYpRi9GLEYvKUkkRHB4R0YqI0YlRitGJUYl

We begin with U(x,0), for instance at x=2 Ix; Ixfinal;Deltapx;

KiQpLCgqKCIiIyIiIkkidEc2IkYnLUkiVUdGKTYkSSJ4R0YpIiIhRidGJyomRiZGJ0YtRidGJyokRighIiJGMUYmRic=

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieEdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweEdGKEYvRiVGJSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRixGKUYzRilGKQ==

LCgiIiJGIyoqI0YjIiIlRiNJIlZHNiJGIywmKiQpRiciIiRGI0YjRiZGI0YjSSJ4R0YoRiMhIiIqKEYlRiMpRiciIiNGIylGLUYxRiNGIw==

We write II for the invariant I(x), and first eliminate the square root involving x res1:=numer(factor(resultant(II-subs(sqrt(Dpx)=sDpx,sqrt(delta)=sdelta,x=2,Ixfinal),subs(x=2,sDpx^2-Deltapx),sDpx))): nops(res1); degree(res1,II); indets(res1)

IiNJ

IiIj

PCVJI0lJRzYiSSJWR0YkSSdzZGVsdGFHRiQ=

Elimination of the other square root res2:=factor(resultant(res1,sdelta^2-1+V^3,sdelta)): nops(res2);

IiIj

op(1,res2);

KiQpSSJWRzYiIiIkIiIi

nops(op(2,res2));indets(res2)

IiNc

PCRJI0lJRzYiSSJWR0Yk

A quartic equation for I(2) over Q(V) res2a:=op(2,res2): degree(res2a,II);

IiIl

Now we move from I to U Ix

KiQpLCgqKCIiIyIiIkkidEc2IkYnLUkiVUdGKTYkSSJ4R0YpIiIhRidGJyomRiZGJ0YtRidGJyokRighIiJGMUYmRic=

Degree 8 for U(2,0) over Q(V) algU2V:=numer(factor(subs(II=(2*t*Ux+2*2-1/t)^2,t=tV,res2a))):nops(algU2V); degree(algU2V,Ux);

IiQ2Iw==

IiIp

Finally, elimination of V algU2:=factor(resultant(algU2V,algV,V)): nops(factor(%));degree(%%,Ux);

IiQ/JA==

IiND

Let us check n:=20: series(subs(Ux=subs(x=2,y=0,User(n)),algU2),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

I(x) and J(x) satisfy the same equation over Q(t,x), hence of degree 12. Now D(y), at y=2 as well res3:=subs(II=JJ, res2a): Jy;

KiYsJiokKSwmKiZJInRHNiIiIiJJInlHRilGKiEiIkYqRioiIiNGKkYqKigiIiVGKilGKEYtRipGK0YsRixGKiksJiomRitGKi1JI0RER0YpNiNGK0YqRioqJEYoRixGKkYtRio=

res4:=numer(factor(subs(JJ=subs(y=2,Delta)*(2*Dy+1/t)^2,t=tV,res3))): nops(%);

IiIj

op(1,res4);

KiQpLCYqJClJIlZHNiIiIiMiIiJGKkYpISIiRilGKg==

nops(op(2,res4));

IiQlRw==

res4a:=op(2,res4): degree(%,Dy);

IiIp

algD2V:=res4a: Elimination of V algDy:=(factor(resultant(algD2V,algV,V))): nops(%); degree(%%,Dy);

IiRAJg==

IiND

Let us check n:=20: series(subs(Dy=subs(y=2,Dser(n)),algDy),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

The degree of C(x,y) is at most 96 eqC;

LCoqJiwmIiIiRiUqJkkidEc2IkYlLCgqJEkieEdGKCEiIkYlKiRJInlHRihGLEYlKiZGK0YlRi5GJUYlRiVGLEYlLUkiQ0dGKDYkRitGLkYlRixGJUYlKihGJ0YlRi5GLC1JI0NtR0YoNiNGKkYlRiwqKEYnRiVGK0YsLUY1NiNGLUYlRiw=

It suffices to determine the degree of xCm(1/x)+yCm(1/y), or equivalently, of U(x,0)+ U(y,0). We do so at x=2 and y=3 We reuse the result obtained for U(2,0) and perform a similar calculation for U(3,0) Elimination of the first square root res1:=numer(factor(resultant(II-subs(sqrt(Dpx)=sDpx,sqrt(delta)=sdelta,x=3,Ixfinal),subs(x=3,sDpx^2-Deltapx),sDpx))): nops(res1); degree(res1,II);

IiNJ

IiIj

Elimination of the other square root res2:=factor(resultant(res1,sdelta^2-1+V^3,sdelta)): nops(res2);

IiIj

op(1,res2);

KiQpSSJWRzYiIiIkIiIi

nops(op(2,res2));

IiNc

res2a:=op(2,res2): degree(res2a,II);

IiIl

Now we move from I to U algU3V:=numer(factor(subs(II=(2*t*Ux+2*3-1/t)^2,t=tV,res2a))):nops(algU3V); degree(algU3V,Ux);

IiQ2Iw==

IiIp

Finally, elimination of V algU3:=factor(resultant(algU3V,algV,V)): nops(factor(%));degree(%%,Ux);

IiQ/JA==

IiND

Let us check n:=20: series(subs(Ux=subs(x=3,y=0,User(n)),algU3),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

Now the sum U(2,0)+U(3,0) indets(algU2V); indets(algU3V);

PCRJI1V4RzYiSSJWR0Yk

PCRJI1V4RzYiSSJWR0Yk

res1:=numer(factor(resultant(sum-Ux-Uy,algU2V,Ux))): nops(res1);

IiQ6KA==

res:=numer(factor(resultant(res1,subs(Ux=Uy,algU3V),Uy))): nops(res);

IiIl

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

NyYiIiIiIiMiJW9KIiVjSg==

NyY8IjwjSSJWRzYiPCRGJUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmRic=

map(degree,[seq(op(i,res),i=1..nops(res))],sum);

NyYiIiFGIyIjS0Yk

Let us check which factor is correct n:=6: series(subs(sum=subs(x=2,y=0,User(n))+subs(x=3,y=0,User(n)),V=Vser(n),op(3,res)),t,n); series(subs(sum=subs(x=2,y=0,User(n))+subs(x=3,y=0,User(n)),V=Vser(n),op(4,res)),t,n);

KzFJInRHNiIiQ3dQJUdaIjNcVzclR11veE5QTCIiIiEhRT9mQSNlP1xVKnAoSEg0L21vbyo9IiIiIkdvXlRALnlxSld2MHkyMSpmOmtJIiIiIyFINyRIM0ReKjNeIlFANSN5KXBsdzotZSIiJCJKKSllJSl6MT94LikzKVFNPHBxMUE/aSc9IiIlIUtbNz0raypSbDM2V14zISoqKSlvKkdja2klIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIic=

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

algsum23V:=op(4,res): res:=factor(resultant(algsum23V,algV,V)): nops(res);

IiVUWg==

degree(res,sum);

IiMnKg==

Jy; Jyfinal;

KiYsJiokKSwmKiZJInRHNiIiIiJJInlHRilGKiEiIkYqRioiIiNGKkYqKigiIiVGKilGKEYtRipGK0YsRixGKiksJiomRitGKi1JI0RER0YpNiNGK0YqRioqJEYoRixGKkYtRio=

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieUdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweUdGKEYvRiVGKSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRixGKUYzRilGKQ==

Taking periodicities into account, we introduce a modified version of J(y), namely V^2 J(y/V)/Delta(y/V)=(yD(y/V)+V/t)^2, which is a series in t^3, or in V^3:=Vt ("t" like "three" or "trois") Jymod:=factor(expand(subs(Dpy=subs(x=y,Deltapx),t=tV,y=y/V,V=Vt^(1/3),delta=1-Vt,V^2*Jyfinal/Delta)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZKeW1vZEdGKCwkKiopLCZJI1Z0R0YoIiIiIiIjRjNGNEYzLDgqKkY0RjNGMkYzSSJ5R0YoRjMpLCZGM0YzRjIhIiIjRjNGNEYzRjMqKEYyRjNGN0YzKSwqKiZGMkYzRjdGM0Y6KiQpRjdGNEYzRjMqJiIiJUYzRjdGM0Y6RkNGM0Y7RjNGMyooRjRGM0ZBRjNGPUYzRjoqKEY0RjNGQUYzRjJGM0YzKiZGNEYzKUY3IiIkRjNGOiooRjRGM0Y3RjNGOEYzRjoqJkYyRjNGPUYzRjMqKEZDRjNGN0YzRjJGM0Y6KiZGQ0YzRkFGM0YzKiZGNEYzRjJGM0YzKiZGNEYzRjdGM0Y6RjMsKkY/RjNGQEY6RkJGM0ZDRjpGOiwmRjdGOkYyRjNGOkY6NyNGLg==

factor(series(Jymod,y,2))

KylJInlHNiIqJCksJkkjVnRHRiQiIiIiIiNGKUYqRikiIiEsJCoqI0YpIiM7RilGJkYpLCwqKCIiKUYpRihGKSksJkYpRilGKCEiIiNGKUYqRilGKSomIiIkRikpRihGKkYpRikqJkYyRilGM0YpRjUqJiIiJUYpRihGKUYpRjJGKUYpRihGNUYpRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKUYq

Expression of the paper (in x rather than y...) exprpap:=2*(2+Vt)^2/(x-Vt)/(4-4*x-x*Vt+x^2)*(x*(1-Vt)^(3/2)+(1-x)^2*(x-Vt)-(Vt+x*Vt-2*x^2)*sqrt(1-(4+Vt)/4*x+x^2/4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHBycGFwR0YoLCQqLCIiIyIiIiksJkkjVnRHRihGMUYwRjFGMEYxLCZJInhHRihGMUY0ISIiRjcsKiomRjRGMUY2RjFGNyokKUY2RjBGMUYxKiYiIiVGMUY2RjFGN0Y9RjFGNywoKiZGNkYxKSwmRjFGMUY0RjcjIiIkRjBGMUYxKiYpLCZGMUYxRjZGN0YwRjFGNUYxRjEqKCNGMUYwRjEsKEY5RjEqJkYwRjFGO0YxRjdGNEYxRjEpRjhGSEYxRjdGMUYxNyNGLg==

factor(Jymod-subs(x=y,exprpap))

IiIh

factor(subs(y=y/V,t=tV,Delta))

KiosKiomKUkiVkc2IiIiJCIiIkkieUdGJ0YpRikqJClGKiIiI0YpISIiKiYiIiVGKUYqRilGKUYwRi5GKSwmKiRGJUYpRilGKkYuRikpLCZGMkYpRi1GKUYtRi5GKkYu

We need to compute in the extension of Q(t,V) generated by sqrt(1-V^3). We still denote V^3=Vt. There is a rational parametrization for this extension algWVt:=Vt-4*W*(1-W);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdXVnRHRigsJkkjVnRHRigiIiIqKCIiJUYwSSJXR0YoRjAsJkYwRjBGMyEiIkYwRjU3I0Yu

VtW:=solve(%,Vt);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWdFdHRigsJCooIiIlIiIiSSJXR0YoRjEsJkYxISIiRjJGMUYxRjQ3I0Yu

Wser:=proc(n) op(2,algeqtoseries(algWVt,Vt,W,n,true)): end: Wser(4);

Ky1JI1Z0RzYiIyIiIiIiJUYmI0YmIiM7IiIjI0YmIiNLIiIkIyIiJiIkYyNGJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmRi8=

factor([solve(algWVt,W)]);

NyQsJiMiIiIiIiNGJSomRiRGJSksJkYlRiVJI1Z0RzYiISIiRiRGJUYlLCZGJEYlRidGLA==

Back to the coefficients D_i subs(DD(y)=add(DD[i]*y^i,i=0..6), y=y/V,t=tV,V^2*Jy/Delta);

KiYpSSJWRzYiIiIjIiIiKSwmKihGJCEiIkkieUdGJUYnLDAqKCZJI0RER0YlNiMiIidGJylGJEYyRispRixGMkYnRicqKCZGMDYjIiImRicpRiRGOEYrKUYsRjhGJ0YnKigmRjA2IyIiJUYnKUYkRj5GKylGLEY+RidGJyooJkYwNiMiIiRGJylGJEZERispRixGREYnRicqKCZGMDYjRiZGJ0YjRispRixGJkYnRicqKCZGMDYjRidGJ0YkRitGLEYnRicmRjA2IyIiIUYnRidGJyomRiRGKywmKiRGRUYnRidGJkYnRidGJ0YmRic=

serD:=subs(csgn(V^3 + 2)=1,simplify(series(sqrt(%),y,5),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJER0YoKy9JInlHRigsJiokKUkiVkdGKCIiJCIiIkY1IiIjRjUiIiEmSSNEREdGKDYjRjdGNSomJkY5NiNGNUY1RjMhIiJGNiomJkY5NiNGNkY1KUYzRjZGPkY0KiYmRjk2I0Y0RjVGMkY+IiIlLUkiT0dGJUY9IiImNyNGLg==

ser:=subs(csgn(Vt + 2)=1,simplify(factor(series(sqrt(Jymod),y,3)),symbolic));

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

The value of D0 D0V:=coeff(%,y,1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMFZHRigsJCoqIyIiIiIjS0YxLCZJI1Z0R0YoRjEiIiNGMUYxLCwqKCIiKUYxRjRGMSksJkYxRjFGNCEiIiNGMUY1RjFGMSomIiIkRjEpRjRGNUYxRjEqJkY4RjFGOUYxRjsqJiIiJUYxRjRGMUYxRjhGMUYxRjRGO0YxNyNGLg==

In terms of W D0W:=factor(simplify(factor(subs(Vt=VtW,D0V))) assuming 0<W assuming W<1/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMFdHRigsJCoqIyIiIiIiJUYxLCoqJiIiJEYxKUkiV0dGKEY1RjFGMSomIiIjRjEpRjdGOUYxISIiKiZGMkYxRjdGMUY7RjJGMUYxLChGOEYxKiZGOUYxRjdGMUY7RjFGO0YxLCZGMUY7RjdGMUY7RjE3I0Yu

Let us check n:=16: series(subs(Vt=V^3,V=Vser(n),D0V),t,n);

Ky9JInRHNiIiIiIiIiEiIiUiIiQiI1kiIiciJDEoIiIqIiZzQyIiIzctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjOg==

n:=16: series(subs(W=Wser(n),Vt=Vser(n)^3,D0W),t,n);

KzFJInRHNiIiIiIiIiEiIiUiIiQiI1kiIiciJDEoIiIqIiZzQyIiIzciJ0snUiMiIzotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjPQ==

subs(y=0,Dser(n));

LC4qJiInSydSIyIiIilJInRHNiIiIzpGJUYlKiYiJnNDIkYlKUYnIiM3RiVGJSomIiQxKEYlKUYnIiIqRiVGJSomIiNZRiUpRiciIidGJUYlKiYiIiVGJSlGJyIiJEYlRiVGJUYl

This gives also C01 C01W:=factor((D0W-1)/2/t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVDMDFXR0YoLCQqLCMiIiIiIilGMUkiV0dGKEYxLCwqJiIiJ0YxKUYzIiIlRjFGMSomIiM1RjEpRjMiIiRGMSEiIiomIiIoRjEpRjMiIiNGMUY9KiYiIz1GMUYzRjFGMUYyRj1GMSwmRjFGPUYzRjFGPUkidEdGKEY9RjE3I0Yu

Let us check n:=16: series(subs(W=Wser(n),Vt=Vser(n)^3,C01W),t,n);

Ky9JInRHNiIiIiNGJSIjQiIiJiIkYCQiIikiJU9pIiM2Iic7KT4iIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM8

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

LCwqJiInOyk+IiIiIilJInRHNiIiIzlGJUYlKiYiJU9pRiUpRiciIzZGJUYlKiYiJGAkRiUpRiciIilGJUYlKiYiI0JGJSlGJyIiJkYlRiUqJiIiI0YlKUYnRjdGJUYl

For D1 now serD

Ky9JInlHNiIsJiokKUkiVkdGJCIiJCIiIkYqIiIjRioiIiEmSSNEREdGJDYjRixGKiomJkYuNiNGKkYqRighIiJGKyomJkYuNiNGK0YqKUYoRitGM0YpKiYmRi42I0YpRipGJ0YzIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYyIiIm

D1V:=V*coeff(ser,y,2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMVZHRigsJCosIyIiIiIlWz9GMUkiVkdGKEYxLCZJI1Z0R0YoRjEiIiNGMUYxLDQqKCIjISlGMSksJkYxRjFGNSEiIiNGMUY2RjEpRjUiIiRGMUYxKiYiI05GMSlGNSIiJUYxRjEqKCIkbyRGMUY6RjEpRjVGNkYxRjEqJiIkKyNGMUY+RjFGMSooIiNrRjFGNUYxRjpGMUY8KiZGSkYxRkZGMUYxKiYiJCVRRjFGOkYxRjwqJiIkRyJGMUY1RjFGPEZNRjFGMUZGRjxGMTcjRi4=

In terms of W D1W:=factor(simplify(factor(subs(Vt=VtW,D1V))) assuming 0<W assuming W<1/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMVdHRigsJCosIyIiIiIja0YxLDAqJiIjTkYxKUkiV0dGKCIiJ0YxRjEqJiIkKyJGMSlGNyIiJkYxISIiKiYiIz9GMSlGNyIiJUYxRjEqJiIkVyJGMSlGNyIiJEYxRjEqJiIjJypGMSlGNyIiI0YxRj0qJiIjS0YxRjdGMUY9RktGMUYxLCgqJkZJRjFGSEYxRjEqJkZJRjFGN0YxRj1GMUY9RjFJIlZHRihGMSksJkYxRj1GN0YxRklGPUY9NyNGLg==

Let us check n:=20: series(subs(Vt=V^3,V=Vser(n),D1V),t,n);

Ky9JInRHNiIiIiJGJSIjNSIiJSIkWyIiIigiJXNERiYiJmkqWyIjOC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM7

n:=16: series(subs(W=Wser(n),Vt=Vser(n)^3,V=Vser(n),D1W),t,n);

Ky9JInRHNiIiIiJGJSIjNSIiJSIkWyIiIigiJXNERiYiJmkqWyIjOC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM7

coeff(Dser(n),y,1);

LC4qJiInZyEqKSoiIiIpSSJ0RzYiIiM7RiVGJSomIiZpKltGJSlGJyIjOEYlRiUqJiIlc0RGJSlGJyIjNUYlRiUqJiIkWyJGJSlGJyIiKEYlRiUqJkYxRiUpRiciIiVGJUYlRidGJQ==

Value of t^2 C_{1,1}=t^2 D1 factor(subs(V=Vt^(1/3),Vt=VtW,tV^2*D1W))

LCQqLCMiIiIiI2tGJSwwKiYiI05GJSlJIldHNiIiIidGJUYlKiYiJCsiRiUpRisiIiZGJSEiIiomIiM/RiUpRisiIiVGJUYlKiYiJFciRiUpRisiIiRGJUYlKiYiIycqRiUpRisiIiNGJUYyKiYiI0tGJUYrRiVGMkZARiVGJUYrRiUsKComRj5GJUY9RiVGJSomRj5GJUYrRiVGMkYlRjJGMiwmRiVGMkYrRiVGMkYl

Ix; Ixfinal;

KiQpLCgqKCIiIyIiIkkidEc2IkYnLUkiVUdGKTYkSSJ4R0YpIiIhRidGJyomRiZGJ0YtRidGJyokRighIiJGMUYmRic=

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieEdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweEdGKEYvRiVGJSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRjNGKUYsRilGKQ==

Deltapx;

LCgiIiJGIyoqI0YjIiIlRiNJIlZHNiJGIywmKiQpRiciIiRGI0YjRiZGI0YjSSJ4R0YoRiMhIiIqKEYlRiMpRiciIiNGIylGLUYxRiNGIw==

Taking periodicities into account, we introduce a variant of I(x), which is a series in V^3. Namely, V^2 I(x/V) subs(x=x/V,V^2*Ix);

KiYpSSJWRzYiIiIjIiIiKSwoKihGJkYnSSJ0R0YlRictSSJVR0YlNiQqJkkieEdGJUYnRiQhIiIiIiFGJ0YnKihGJkYnRjBGJ0YkRjFGJyokRitGMUYxRiZGJw==

Ixmod:=subs(Dpx=Deltapx,x=x/V,V^2*Ixfinal);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZJeG1vZEdGKComKUkiVkdGKCIiIyIiIiwoKixGMUYyLCZGMEYyRjIhIiJGMiwoKiRGL0YyRjJGMEYyRjJGMkYyRi9GNilJJmRlbHRhR0YoI0YyRjFGMkY2KipGMUYyLCgqJkYwRjJJInhHRihGMkYyKihGMUYyKUY/RjFGMkYvRjZGNkYwRjJGMkY/RjYpLChGMkYyKigjRjIiIiVGMiwmKiQpRjAiIiRGMkYyRkZGMkYyRj9GMkY2KiZGRUYyRkFGMkYyRjtGMkYyKixGMUYyLCZGOEYyKiZGP0YyRjBGNkY2RjIpLCZGP0YyRjJGNkYxRjJGP0Y2RjBGNkY2RjI3I0Yu

Ixmod:=subs(V=Vt^(1/3),delta=1-Vt,factor(Ixmod))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZJeG1vZEdGKCwkKiYsOCoqIiIjIiIiKSwmRjNGM0kjVnRHRighIiIjRjNGMkYzRjZGM0kieEdGKEYzRjMqKCksKiomRjZGM0Y5RjNGNyokKUY5RjJGM0YzKiYiIiVGM0Y5RjNGN0ZBRjNGOEYzRjZGM0Y5RjNGNyooRjJGM0Y2RjNGP0YzRjMqJkY7RjNGNkYzRjcqKEZBRjNGNkYzRjlGM0Y3KihGMkYzRjtGM0Y/RjNGMyomRjJGM0Y2RjNGMyomRjJGMylGOSIiJEYzRjcqKEYyRjNGNEYzRjlGM0Y3KiZGQUYzRj9GM0YzKiZGMkYzRjlGM0Y3RjNGOUY3Rjc3I0Yu

The expression of the paper exprpap:=2/x*(x*(1-Vt)^(3/2)+(1-x)^2*(x-Vt)+(Vt+x*Vt-2*x^2)*sqrt(1-(4+Vt)/4*x+x^2/4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHBycGFwR0YoLCQqKCIiIyIiIkkieEdGKCEiIiwoKiZGMkYxKSwmRjFGMUkjVnRHRihGMyMiIiRGMEYxRjEqJiksJkYxRjFGMkYzRjBGMSwmRjJGMUY4RjNGMUYxKigjRjFGMEYxLCgqJkY4RjFGMkYxRjEqJkYwRjEpRjJGMEYxRjNGOEYxRjEpLCpGQkYzKiRGREYxRjEqJiIiJUYxRjJGMUYzRklGMUZARjFGMUYxRjE3I0Yu

factor(Ixmod-exprpap)

IiIh

factor(series(Ixmod,x,6));

Ky9JInhHNiIsLCooIiIjIiIiSSNWdEdGJEYoKSwmRihGKEYpISIiI0YoRidGKEYsKiYiIiZGKEYpRihGKComI0YoIiIlRigpRilGJ0YoRiwqJkYnRihGKkYoRihGJ0YoIiIhLCQqJiNGKCIja0YoKSwmRilGKCIiKUYoIiIkRihGLEYoLCQqKCNGKCIkNyZGKCwmRilGKEYyRixGKEY6RihGLEYnLCQqKiNGKCImJVE7RihGKUYoLCZGLkYoRjxGLEYoRjpGKEYsRj0sJCoqI0YoIidzNThGKEYpRigsKComIiIoRihGM0YoRigqJiIjN0YoRilGKEYoIiNLRixGKEY6RihGLEYyLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGLw==

The constant term coeff(%,x,0);

LCwqKCIiIyIiIkkjVnRHNiJGJSksJkYlRiVGJiEiIiNGJUYkRiVGKiomIiImRiVGJkYlRiUqJiNGJSIiJUYlKUYmRiRGJUYqKiZGJEYlRihGJUYlRiRGJQ==

factor(subs(sqrt(1-Vt)=0,%))

LCgiIiMiIiIqJiIiJkYkSSNWdEc2IkYkRiQqJiNGJCIiJUYkKUYnRiNGJCEiIg==

Introduce W factor(subs(sqrt(1-Vt)=1-2*W,Vt=VtW,%%))

LCQqKCIiJSIiIiwmRiUhIiJJIldHNiJGJUYlKSwmRihGJUYlRiUiIiRGJUYn

IxmodW:=factor(subs(sqrt(1-Vt)=1-2*W,Vt=VtW,Ixmod));

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

factor(series(IxmodW,x,5));

Ky1JInhHNiIsJCooIiIlIiIiLCZGKCEiIkkiV0dGJEYoRigpLCZGK0YoRihGKCIiJEYoRioiIiEqJkYsRigpLCZGK0YoIiIjRipGLkYoRigsJCoqI0YoRjNGKCwoKiQpRitGM0YoRihGK0YqRihGKEYoRixGKEYxRihGKkYzLCQqLiNGKCIjO0YoRitGKEYpRigsKComIiImRihGOUYoRigqJkZARihGK0YoRipGM0YoRihGLEYoRjFGKEYoRi4tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYn

We want the constant term (in x and t) to be -2, not 2 serroot:=factor(simplify(factor(series(-sqrt(%),x)) assuming -1<W assuming W<1))

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

The coefficients are of the form sqrt(1-W^2) Pol(W, 1/(1-W^2)) (experimentally, it is even Pol(W, 1/(1-W))). We now need sqrt(1-W^2), and thus introduce the series Z algZW:=numer(W-2*Z/(1+Z^2));collect(%,W,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdaV0dGKCwoKiZJIldHRigiIiIpSSJaR0YoIiIjRjFGMUYwRjEqJkY0RjFGM0YxISIiNyNGLg==

LCYqJkkiV0c2IiIiIiwmKiQpSSJaR0YlIiIjRiZGJkYmRiZGJkYmKiZGK0YmRipGJiEiIg==

WZ:=factor(solve(%,W))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNXWkdGKCwkKigiIiMiIiJJIlpHRihGMSwmKiQpRjJGMEYxRjFGMUYxISIiRjE3I0Yu

ZWser:=proc(n) op(algeqtoseries(algZW,W,Z,n,true)): end: ZWser(5);

KytJIldHNiIjIiIiIiIjRiYjRiYiIikiIiQjRiYiIzsiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

simplify(factor(subs(W=WZ,sqrt(1-W^2)))) assuming Z>0 assuming Z<1

KigsJkkiWkc2IiIiIkYmRiZGJiwmKiQpRiQiIiNGJkYmRiZGJiEiIiwmRiZGJkYkRitGJg==

algZVt:=factor(resultant(algZW,algWVt,W));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdaVnRHRigsLiomSSNWdEdGKCIiIilJIlpHRigiIiVGMUYxKigiIiNGMUYwRjEpRjNGNkYxRjEqJiIiKUYxKUYzIiIkRjEhIiIqJiIjO0YxRjdGMUYxRjBGMSomRjlGMUYzRjFGPDcjRi4=

algZ:=collect(factor(resultant(subs(Vt=V^3,%),algV,V)),t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCYqKCIiKSIiIiksLCokKUkiWkdGKCIiJUYxRjEqJkY3RjEpRjYiIiRGMUYxKiYiIidGMSlGNiIiI0YxISIiKiZGN0YxRjZGMUYxRjFGMUY6RjEpSSJ0R0YoRjpGMUYxKipGMEYxRjZGMSksJkY2RjFGMUY/Rj5GMSksJiokRj1GMUYxRjFGMUY3RjFGPzcjRi4=

tcubeZ:=factor(solve(%,t^3));

Warning, solving for expressions other than names or functions is not recommended.

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0Y3ViZVpHRigqKkkiWkdGKCIiIiksJkYvRjBGMCEiIiIiI0YwKSwmKiQpRi9GNEYwRjBGMEYwIiIlRjApLCwqJClGL0Y5RjBGMComRjlGMClGLyIiJEYwRjAqJiIiJ0YwRjhGMEYzKiZGOUYwRi9GMEYwRjBGMEZARjM3I0Yu

serrootZ:=factor(subs(sqrt((Z-1)^2*(Z+1)^2/(Z^2+1)^2)=(1-Z)*(1+Z)/(1+Z^2),factor(subs(W=WZ,serroot))));

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

Back to Cm(x) subs(U(x,0)=add(Cm[i]*x^(i-1),i=1..8),x=x/V,t=tV,V^2*Ix);

KiYpSSJWRzYiIiIjIiIiKSwoKipGJkYnRiRGJywmKiQpRiQiIiRGJ0YnRiZGJyEiIiwyKigmSSNDbUdGJTYjIiIpRicpSSJ4R0YlIiIoRicpRiRGOEYvRicqKCZGMzYjRjhGJylGNyIiJ0YnKUYkRj5GL0YnKigmRjM2I0Y+RicpRjciIiZGJylGJEZERi9GJyooJkYzNiNGREYnKUY3IiIlRicpRiRGSkYvRicqKCZGMzYjRkpGJylGN0YuRidGLUYvRicqKCZGMzYjRi5GJylGN0YmRidGI0YvRicqKCZGMzYjRiZGJ0Y3RidGJEYvRicmRjM2I0YnRidGJ0YnKihGJkYnRjdGJ0YkRi9GJyomRiRGL0YrRidGL0YmRic=

mind the sign: the square root that we want is: subs(U(x,0)=add(Cm[i]*x^(i-1),i=1..8),x=x/V,t=tV,V*Rx);

KiZJIlZHNiIiIiIsKCoqIiIjRiVGI0YlLCYqJClGIyIiJEYlRiVGKEYlISIiLDIqKCZJI0NtR0YkNiMiIilGJSlJInhHRiQiIihGJSlGI0Y2Ri1GJSooJkYxNiNGNkYlKUY1IiInRiUpRiNGPEYtRiUqKCZGMTYjRjxGJSlGNSIiJkYlKUYjRkJGLUYlKigmRjE2I0ZCRiUpRjUiIiVGJSlGI0ZIRi1GJSooJkYxNiNGSEYlKUY1RixGJUYrRi1GJSooJkYxNiNGLEYlKUY1RihGJSlGI0YoRi1GJSooJkYxNiNGKEYlRjVGJUYjRi1GJSZGMTYjRiVGJUYlRiUqKEYoRiVGNUYlRiNGLUYlKiZGI0YtRilGJUYtRiU=

factor(series(%,x,4));

Ky1JInhHNiIsJComLCoqJClJIlZHRiQiIiciIiJGLComIiIlRiwpRioiIiRGLEYsKigiIiNGLClGKkYyRiwmSSNDbUdGJDYjRixGLCEiIkYuRixGLCwmKiRGL0YsRixGMkYsRjdGNyIiISwkKihGMkYsLChGOUYsKiZGKkYsJkY1NiNGMkYsRixGMkYsRixGOEY3RixGLCwkKihGMkYsRjhGNyZGNTYjRjBGLEYsRjIsJCoqRjJGLEYqRjdGOEY3JkY1NiNGLkYsRixGMC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGNkYu

serrootZ;

Ky1JInhHNiIsJCoqIiIjIiIiKSwmSSJaR0YkRihGKEYoIiIkRigsJkYrRihGKCEiIkYoKSwmKiQpRitGJ0YoRihGKEYoRidGLkYoIiIhLCQqLEYnRihGKUYoKSwoRjFGKEYrRi5GKEYoRixGKClGMCIiJUYuRi1GLkYuRigsJCouRilGKEY2RihGK0YoLCwqJClGK0Y5RihGKComIiImRigpRitGLEYoRi4qJiIiKkYoRjJGKEYoKiZGQEYoRitGKEYuRihGKEYoKUYtRixGLilGMCIiJ0YuRi5GJywkKjAjRihGJ0YoRilGKEY2RihGK0YoLDgqJClGKyIjNUYoRigqJkZARigpRitGQ0YoRi4qJkYsRigpRisiIilGKEYoKiYiI01GKClGKyIiKEYoRigqJiIkLyJGKClGK0ZHRihGLiomIiRXIkYoKUYrRkBGKEYoKiZGWUYoRj5GKEYuKiZGVUYoRkFGKEYoKiZGLEYoRjJGKEYoRkRGLkYoRihGKClGLUZARi4pRjBGU0YuRi5GLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRjk=

series(%%-%,x,3);

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

convert(%,polynom): {coeffs(%,x)};

PCUsJiooIiIjIiIiLCYqJClJIlZHNiIiIiRGJkYmRiVGJiEiIiZJI0NtR0YrNiNGLEYmRiYqLiksJkkiWkdGK0YmRiZGJkYsRiYpLCgqJClGNEYlRiZGJkY0Ri1GJkYmRixGJkY0RiYsLCokKUY0IiIlRiZGJiomIiImRiYpRjRGLEYmRi0qJiIiKkYmRjhGJkYmKiZGPkYmRjRGJkYtRiZGJkYmKSwmRjRGJkYmRi1GLEYtKSwmRjdGJkYmRiYiIidGLUYmLCYqKEYlRiYsKEYoRiYqJkYqRiYmRi82I0YlRiZGJkYlRiZGJkYnRi1GJiosRiVGJkYyRiZGNUYmKUZGRjxGLUZERi1GJiwmKiYsKiokKUYqRkdGJkYmKiZGPEYmRilGJkYmKihGJUYmKUYqRiVGJiZGLzYjRiZGJkYtRjxGJkYmRidGLUYtKipGJUYmRjJGJkZERiYpRkZGJUYtRi0=

coCm:=op(factor(solve(%,[Cm[1],Cm[2],Cm[3]])));

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

This is C(-1,0)/t factor(subs(V=Vt^(1/3),Vt=VtW,W=WZ,op(2,coCm[1])/tV));

KiosKiokKUkiWkc2IiIiJCIiIkYpKiZGKEYpKUYmIiIjRilGKSomRihGKUYmRikhIiJGKUYpRikpLCwqJClGJiIiJUYpRikqJkYzRilGJUYpRikqJiIiJ0YpRitGKUYuKiZGM0YpRiZGKUYpRilGKUYsRikpLCZGJkYpRilGLkYsRi4pLCYqJEYrRilGKUYpRilGM0Yu

Cm10overt:=factor(%*tcubeZ/t^3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpDbTEwb3ZlcnRHRigqKiwqKiQpSSJaR0YoIiIkIiIiRjQqJkYzRjQpRjIiIiNGNEY0KiZGM0Y0RjJGNCEiIkY0RjRGNCwsKiQpRjIiIiVGNEY0KiZGPUY0RjFGNEY0KiYiIidGNEY2RjRGOSomRj1GNEYyRjRGNEY0RjRGOUYyRjQpSSJ0R0YoRjNGOTcjRi4=

Cm10timest2:=t^3*Cm10overt;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxDbTEwdGltZXN0MkdGKCooLCoqJClJIlpHRigiIiQiIiJGNComRjNGNClGMiIiI0Y0RjQqJkYzRjRGMkY0ISIiRjRGNEY0LCwqJClGMiIiJUY0RjQqJkY9RjRGMUY0RjQqJiIiJ0Y0RjZGNEY5KiZGPUY0RjJGNEY0RjRGNEY5RjJGNDcjRi4=

Let us check Zser:=proc(n) op(3,algeqtoseries(numer(tcubeZ-t^3),t, Z,n, true)): end: Zser(6);

KylJInRHNiIiIiIiIiQiIzkiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKg==

n:=20: series(subs(Z=Zser(n), Cm10overt),t,n);

KzFJInRHNiIiIiIiIiEiIigiIiQiIyEqIiInIiVcOSIiKiImJ0dFIiM3IidJT14iIzotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjPQ==

expand(coeff(Cmser(n)/t,x,-1));

LDAqJiIpIVFmMCIiIiIpSSJ0RzYiIiM9RiVGJSomIidJT15GJSlGJyIjOkYlRiUqJiImJ0dFRiUpRiciIzdGJUYlKiYiJVw5RiUpRiciIipGJUYlKiYiIyEqRiUpRiciIidGJUYlKiYiIihGJSlGJyIiJEYlRiVGJUYl

algV:=-V+t*(V^3+2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRighIiIqJkkidEdGKCIiIiwmKiQpRi8iIiRGM0YzIiIjRjNGM0YzNyNGLg==

Here, tt stands for t three (t cubed). This is the equation for V^3=Vt algVt:=Vt-tt*(2+Vt)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdWdEdGKCwmSSNWdEdGKCIiIiomSSN0dEdGKEYwKSwmIiIjRjBGL0YwIiIkRjAhIiI3I0Yu

The only singularity is at tt=1/27 factor(discrim(algVt,Vt));

LCQqKCIiJSIiIkkjdHRHNiJGJSwmKiYiI0ZGJUYmRiVGJUYlISIiRiVGKw==

with(gfun):algeqtoseries(subs(tt=1/27*(1-epsilon),algVt),epsilon,Vt,5);;

NyQrL0koZXBzaWxvbkc2IiEiKSIiISNGJiIiJCIiIiMhJGciIiMiKSIiIyMhJSVlJCIlKD0jRikjISZnIkciJiRvPiIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YqIiImLC5GKkYqKiYtSSdSb290T2ZHRjg2IywmKiQpSSNfWkdGOEYuRipGKkYpISIiRiopRiQjRipGLkYqRioqJiNGNUYpRipGJEYqRioqKCMiI04iI2FGKkY/RiopRiQjRilGLkYqRioqJiMiIyEpRi1GKilGJEYuRipGKi1GNzYjKiQpRiQjRjxGLkYqRio=

Vtsing:=subs(RootOf(_V^2-3)=-sqrt(3),op(2,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWdHNpbmdHRigsLiIiIkYvKiYpIiIkI0YvIiIjRi8pSShlcHNpbG9uR0YoRjNGLyEiIiomIyIiJUYyRi9GNkYvRi8qKCMiI04iI2FGL0YxRi8pRjYjRjJGNEYvRjcqJiMiIyEpIiMiKUYvKUY2RjRGL0YvLUkiT0dGJTYjKiQpRjYjIiImRjRGL0YvNyNGLg==

with(plots): implicitplot(algVt,tt=0..1/27,Vt=0..1.5,numpoints=10000);

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series(sqrt(1-Vtsing),epsilon);

LCwqJikiIiQjIiIiIiIlRicpSShlcHNpbG9uRzYiRiZGJ0YnKigjIiIjIiIqRicpRiUjRiVGKEYnKUYqRjFGJyEiIiooRiZGJ0YkRicpRiojIiImRihGJ0YnKigjIiNgIiQnW0YnRjBGJylGKiMiIihGKEYnRjMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YrNiMqJClGKiNGL0YoRidGJw==

Values where Vt=1 subs(Vt=1,algVt)

LCYiIiJGIyomIiNGRiNJI3R0RzYiRiMhIiI=

Now the series W algWVt;

LCZJI1Z0RzYiIiIiKigiIiVGJUkiV0dGJEYlLCZGJUYlRighIiJGJUYq

As a function of Vt, the only singularity is at Vt=1, which happens to be the singular value of Vt in terms of t^3 factor(discrim(%,W));

LCYqJiIjOyIiIkkjVnRHNiJGJSEiIkYkRiU=

op(algeqtoseries(subs(Vt=1-delta,algWVt),delta,W,12));

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKUkmZGVsdGFHRixGI0YkRiQtSSJPR0YpNiMqJClGNSIiJ0YkRiQ=

subs(delta=1-Vtsing,%);

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSwsKiYpIiIkRiNGJClJKGVwc2lsb25HRixGI0YkRiQqJiNGMEY4RiRGOkYkRjMqKCMiI04iI2FGJEY3RiQpRjojRjhGJUYkRiQqJiMiIyEpIiMiKUYkKUY6RiVGJEYzLUkiT0dGKTYjKiQpRjojIiImRiVGJEYzRiNGJEYkLUZJNiMqJClGNSIiJ0YkRiQ=

series(%,epsilon);

LC4jIiIiIiIjRiQqKC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSIiJCNGJEYwRiQpSShlcHNpbG9uR0YsRjZGJEYkKiojRiUiIipGJEYnRiQpRjUjRjVGMEYkKUY4Rj1GJEYzKipGNkYkRidGJEY0RiQpRjgjIiImRjBGJEYkKiojIiNgIiQnW0YkRidGJEY8RiQpRjgjIiIoRjBGJEYzLUkiT0dGKTYjKiQpRjgjRjtGMEYkRiQ=

Recall that W is an increasing function of Vt Wsing:=subs(RootOf(4*_V^2-1)=-1/2,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZXc2luZ0dGKCwuIyIiIiIiI0YwKihGL0YwKSIiJCNGMCIiJUYwKUkoZXBzaWxvbkdGKEY1RjAhIiIqKCNGMCIiKkYwKUY0I0Y0RjZGMClGOEY+RjBGMCooI0YwIiIpRjBGM0YwKUY4IyIiJkY2RjBGOSooIyIjYCIkcypGMEY9RjApRjgjIiIoRjZGMEYwLUkiT0dGJTYjKiQpRjgjRjxGNkYwRjA3I0Yu

series(1-Wsing^2,epsilon);

LDYjIiIkIiIlIiIiKigjRiYiIiNGJilGJCNGJkYlRiYpSShlcHNpbG9uRzYiRitGJkYmKihGK0YmKUYkRihGJilGLUYoRiYhIiIqKCNGJiIiKkYmKUYkRiNGJilGLUYjRiZGMiomI0YmRiRGJkYtRiZGJiooI0YmIiIpRiZGKkYmKUYtIyIiJkYlRiZGJiooIyIjTiIkOyNGJkYwRiYpRi0jRiRGKUYmRjIqKCMiI2AiJHMqRiZGNkYmKUYtIyIiKEYlRiZGMiomIyIjPyIjIilGJilGLUYpRiZGJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRi42IyokKUYtI0Y1RiVGJkYm

Equation of W over Q(t^3) algW:=factor(resultant(algWVt,algVt,Vt));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdXR0YoLDQqKCIjayIiIilJIldHRigiIidGMUkjdHRHRihGMUYxKigiJCM+RjEpRjMiIiZGMUY1RjEhIiIqKCIjJypGMSlGMyIiJUYxRjVGMUYxKigiJEciRjEpRjMiIiRGMUY1RjFGMSooIiNbRjEpRjMiIiNGMUY1RjFGOiomRj5GMUZFRjFGOiooRkRGMUYzRjFGNUYxRjoqJkY+RjFGM0YxRjEqJiIiKUYxRjVGMUY6NyNGLg==

implicitplot(algW,tt=0..1/27,W=0..1,numpoints=10000);

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Finally, the series Z factor(discrim(algZW,Z));

LCQqKCIiJSIiIiwmRiUhIiJJIldHNiJGJUYlLCZGKEYlRiVGJUYlRic=

implicitplot(algZW,W=0..1/2,Z=-2..2);

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noB!#;7$$"2$=&Q$pm8$\%!#<$"27kq%=X3tB!#<7$$"#X!"#$"2K%pyAK9xB!#<7$$"1c!eE3u">X!#;$"1f'pAL%\)Q#!#;7$$"2$*oH1jn*RX!#<$"2&fG=kZ#3S#!#<7$$"2jYUnK%=XX!#<$"2Ou2!=;#RS#!#<7$$"$b%!"$$"0<Dmf"y1C!#:7$$"1-g(fDn6d%!#;$"2#)=F$pkO>C!#<7$$"201Suh>Sf%!#<$"1%fusmyHV#!#;7$$"1*QCahArf%!#;$"1J1<]!H[V#!#;7$$"#Y!"#$"2u?r8KXlV#!#<7$$"1P?+S-0BY!#;$"2[*4qm%4.X#!#<7$$"2/BJSgAyk%!#<$"1HjOqD8lC!#;7$$"21nDG*)\*[Y!#<$"2dr_7(y!eY#!#<7$$"$l%!"$$"2<%)H2*pVmC!#<7$$"2Cd1jQ@[n%!#<$"2G)*fWQC8[#!#<7$$"#Z!"#$"2kVbtDfk\#!#<7$$"14<7/Xm+Z!#;$"1%RNa9fo\#!#;7$$"2c67ibu8q%!#<$"2N1eMZ'G(\#!#<7$$"29uHU.zks%!#<$"283Q0H7C^#!#<7$$"$v%!"$$"1J%\\$[hED!#;7$$"/7B"ykAv%!#9$"2&e^UXR)z_#!#<7$$"1"o'="[tYv%!#;$"2')z\lPS%HD!#<7$$"2%\ir5:-yZ!#<$"2xWS3EuNa#!#<7$$"#[!"#$"2%4*[/]1pb#!#<7$$"0CwY/\P![!#:$"2ZSd7N$=fD!#<7$$"2WqIwU<x![!#<$"1L:kzUfhD!#;7$$"1L,:IrWH[!#;$"1E(\$z8"[d#!#;7$$"$&[!"$$"2uC6j4Pte#!#<7$$"1N&QEg:^&[!#;$"2CO.4Xe/f#!#<7$$"2m!ev&[/0'[!#<$"1oKt#=[Pf#!#;7$$"1"Q*=%>a2)[!#;$"2wSK#QZ71E!#<7$$"#\!"#$"2'*)***G`4zh#!#<7$$"2jg#[]FO1\!#<$"22rw0M5=i#!#<7$$"2P$3O(yKI"\!#<$"2E+De3-fi#!#<7$$"1E3g"*4%>$\!#;$"2%**\%zV:vj#!#<7$$"$&\!"$$"21kx4zE'[E!#<7$$"21$**eq()[d\!#<$"0CS[7SKl#!#:7$$"1K2(e]+`'\!#;$"1Pn"*))f0eE!#;7$$"1;[4)z0I)\!#;$"2&\.M(e%)*oE!#<7$$""&!""$"1D7JC>\zE!#;-%&COLORG6&%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"W6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"Z6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$+(!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%IJ!""-%.BOUNDS_HEIGHTG6#$"%+N!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"G6"

What about the series Z? (will not be needed, in fact) algeqtoseries(subs(W=1/2-kappa,algZW),kappa,Z,6);;

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

solve(_Z^2-4*_Z+1); evalf([%]);

NiQsJiIiIyIiIiokKSIiJCNGJUYkRiVGJSwmRiRGJUYmISIi

NyQkIiszMzBLUCEiKiQiKiM+XHpFRiU=

Zkappasing:=subs(RootOf(_Z^2-4*_Z+1)=2-sqrt(3),op(%%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStaa2FwcGFzaW5nR0YoKzFJJmthcHBhR0YoLCYiIiMiIiIqJCkiIiQjRjJGMUYyISIiIiIhLCYiIiVGMiomIyIiKUY1RjJGNEYyRjdGMiwmRj1GMiomIyIjUyIiKkYyRjRGMkY3RjEsJiIjO0YyKiYjIiRjIyIjRkYyRjRGMkY3RjUsJiIjS0YyKiYjIiVzOSIjIilGMkY0RjJGN0Y6LCYiI2tGMiomIyIlKTMqIiRWI0YyRjRGMkY3IiImLUkiT0dGJTYjRjIiIic3I0Yu

Zsing:=series(subs(kappa=1/2-Wsing,Zkappasing),epsilon);

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

evalf(%);

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Walks ending on the diagonal Recall that we have introduced a modified version of J(y), namely V^2 J(y/V)/Delta(y/V). It is the square of (yD(y/V)+V/t). Here we introduce a variant, namely t^2 J(y/t) /Delta(y/t) factor(subs(y=y/t,t^2*Jy/Delta));

KiQpLCYqJkkieUc2IiIiIi1JI0RER0YnNiMqJkYmRihJInRHRichIiJGKEYoRihGKCIiI0Yo

The same series, in terms of Vt=V^3 Jynew:=factor(subs(V=Vt^(1/3),expand(subs(Dpy=subs(x=y,Deltapx),delta=1-V^3,y=y/t,t=tV,t^2*Jyfinal/Delta))));

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res:=map(factor,series(subs(Vt=Vtsing,Jynew),epsilon,1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsKCooLCwqKCIiJyIiIkkieUdGKEYzKSomLCYqJiIiJEYzRjRGM0YzRjMhIiJGMywmRjhGMyIiJUY6RjMjRjMiIiNGM0YzKiYiIz1GMylGNEY+RjNGMyokRjVGM0YzKiYiIzdGM0Y0RjNGOkY+RjNGM0Y3RjpGO0Y6RjMqLkYyRjNGNEYzKUY5I0Y5RjxGMylJKGVwc2lsb25HRihGR0YzKUY3Rj5GOkY7RjpGMy1JIk9HRiU2I0ZJRjM3I0Yu

Let us examine the first two coefficients d0:=subs(delta=(3*y-1)*(3*y-4),collect(subs(sqrt((3*y-1)*(3*y-4))=sqrt(delta),op(1,%)),delta,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKiosJiomIiInIiIiSSJ5R0YoRjNGM0YzRjNGMywmKiYiIiRGM0Y0RjNGM0YzISIiRjgsJkY2RjMiIiVGOEY4KSomRjVGM0Y5RjMjRjMiIiNGM0YzKihGPkYzRjVGM0Y5RjhGMzcjRi4=

d1:=subs(epsilon=1,op(2,res));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMUdGKCwkKiwiIiciIiJJInlHRihGMSkiIiQjRjQiIiVGMSksJiomRjRGMUYyRjFGMUYxISIiIiIjRjosJkY5RjFGNkY6RjpGMTcjRi4=

We need to take the square root, and divide by y. First, consider the reciprocal of d0 collect(subs(sqrt((3*y-1)*(3*y-4))=sqrt(delta),factor(rationalize(1/d0))),delta,factor);

LCYqLCMiIiIiIyIpRiUsJiomIiInRiVJInlHNiJGJUYlRiVGJUYlLCYqJiIiJEYlRipGJUYlIiIlISIiRiVGKkYwKUkmZGVsdGFHRisjRiUiIiNGJUYwKiojRjRGJkYlKSwmRi1GJUYlRjBGNEYlRixGJUYqRjBGJQ==

c1:=d1*sqrt(%)/2/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCwkKiwjIiIiIiIkRjEpRjIjRjIiIiVGMSksJiomRjJGMUkieUdGKEYxRjFGMSEiIiIiI0Y6LCZGOEYxRjVGOkY6KSwmKiosJiomIiInRjFGOUYxRjFGMUYxRjFGPEYxRjlGOilJJmRlbHRhR0YoI0YxRjtGMUY6KipGO0YxRjZGMUY8RjFGOUY6RjFGRUYxRjE3I0Yu

series(subs(delta=(1-3*y)*(4-3*y),c1),y,4);

Ky1JInlHNiIsJComIyIiJCIiJSIiIilGKEYnRiohIiIiIiEsJComIyIkOCYiJEciRipGK0YqRixGKiwkKiYjIidQZTgiJSM+KUYqRitGKkYsIiIjLCQqJiMiKUAwSTsiJ1dARUYqRitGKkYsRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYp

collect(c1^2,delta,factor);

LCYqMCMiIiIiIiRGJSlGJiNGJSIiI0YlKSwmKiZGJkYlSSJ5RzYiRiVGJUYlISIiIiIlRi8sJkYsRiVGMEYvRi8sJiomIiInRiVGLUYlRiVGJUYlRiVGLUYvKUkmZGVsdGFHRi5GKEYlRi8qLCNGKUYmRiVGJ0YlKUYrRilGL0YxRi9GLUYvRiU=

Paper version c1pap:=-1/3^(1/4)/(1-3*y)/sqrt(y*(4-3*y))*sqrt((1+6*y)/(1-3*y)*sqrt((4-3*y)/(1-3*y))-2);

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series(subs(delta=(1-3*y)*(4-3*y),c1)-c1pap,y);

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

Hdexpl:=-3^(7/4)*subs(y=y/3,c1pap);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZGV4cGxHRigsJCoqIiIqIiIiLCZGMUYxSSJ5R0YoISIiRjQpKiZGM0YxLCYiIiVGMUYzRjRGMSNGMSIiI0Y0KSwmKigsJiomRjpGMUYzRjFGMUYxRjFGMUYyRjQpKiZGN0YxRjJGNEY5RjFGMUY6RjRGOUYxRjE3I0Yu

series(Hdexpl,y)

KzFJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLCQqJiMiJVI6IiRHIkYqRitGKkYqRiosJComIyInUGU4IiUjPilGKkYrRipGKkYuLCQqJiMiKDJOViYiJ1dARUYqRitGKkYqRiwsJComIyIqMCYpUkQpIilLV2JMRipGK0YqRipGKSwkKiYjIixgQ3FhLiQiK0M9dXQ1RipGK0YqRioiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiIiJw==

Let us check this numerically First, at (0,0) N:=100: i:=0: ser:=coeff(Dser(N), y,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=9..12,style=point);

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NiI=

series(Hdexpl,y,1);evalf(%);

KydJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGKg==

KydJInlHNiIkIismSE0icDYhIikiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLw==

At (1,1) N:=120: i:=1: ser:=coeff(Dser(N), y,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=18..21,style=point);

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Ig==

series(Hdexpl,y,2);evalf(%);

KylJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLCQqJiMiJVI6IiRHIkYqRitGKkYqRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYu

KylJInlHNiIkIismSE0icDYhIikiIiEkIitrL18jMyNGJyIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YrIiIj

Walks ending on the boundary (negative x-axis) Ixfinal ;

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqLEYkRiUsKComRixGJUkieEdGKEYlRiUqJkYkRiUpRjNGJEYlRilGJ0YlRiVGJ0YpRjNGKSlJJERweEdGKEYvRiVGJSosRiRGJSwmRitGJUYzRilGJSksJiomRidGJUYzRiVGJUYlRilGJEYlRjNGKUYsRilGKQ==

We consider this variant of I(x), which is a series in t^3. It is x^2t^4 I(x/t) factor(subs(U(x,0)=1/x*Cm(x),x=x/t,x^2/t^2*Ix));

KiYpLCgqKCIiIyIiIilJInRHNiIiIiRGJy1JI0NtR0YqNiMqJkkieEdGKkYnRikhIiJGJ0YnKiZGJkYnKUYwRiZGJ0YnRjBGMUYmRicpRikiIidGMQ==

In terms of V^3 factor(subs(V=Vt^(1/3),expand(subs(Dpx=Deltapx, delta=1-V^3,x=x/t,t=tV,1/t^2*x^2*Ixfinal))));

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

res:=map(factor,series(subs(Vt=Vtsing,%),epsilon,1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsKCoqIiNGIiIiLCYqJiIiJEYxSSJ4R0YoRjFGMUYxISIiRjEsLCooIiInRjFGNUYxKSomRjJGMSwmRjNGMSIiJUY2RjEjRjEiIiNGMUYxKiYiIz1GMSlGNUY/RjFGNiokRjpGMUYxKiYiIzdGMUY1RjFGMUY/RjZGMUY1RjFGNioqIiRpIkYxKUY0I0Y0Rj1GMUZCRjEpSShlcHNpbG9uR0YoRklGMUYxLUkiT0dGJTYjRktGMTcjRi4=

d0:=collect(subs(sqrt((3*x-1)*(3*x-4))=sqrt(delta),op(1,res)),delta,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKiwiI0YiIiIsJiomIiIkRjFJInhHRihGMUYxRjEhIiJGMSwmKiYiIidGMUY1RjFGMUYxRjFGMUY1RjEpSSZkZWx0YUdGKCNGMSIiI0YxRjYqKCIjYUYxKUYyRjRGMUY1RjFGMTcjRi4=

d1:=subs(epsilon=1,op(2,res));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMUdGKCwkKigiJGkiIiIiKSIiJCNGMyIiJUYxKUkieEdGKCIiI0YxRjE3I0Yu

Careful when taking the square root series(subs(delta=(3*x-1)*(3*x-4),d0),x,3)

KydJInhHNiIjIiUoPSMiIiUiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiQ=

Cm10overt

KiosKiokKUkiWkc2IiIiJCIiIkYpKiZGKEYpKUYmIiIjRilGKSomRihGKUYmRikhIiJGKUYpRiksLCokKUYmIiIlRilGKSomRjJGKUYlRilGKSomIiInRilGK0YpRi4qJkYyRilGJkYpRilGKUYpRi5GJkYpKUkidEdGJ0YoRi4=

Value of Z at 1/27? [solve(algZW,Z)]

NyQqJiwmIiIiRiUqJCksJiokKUkiV0c2IiIiI0YlISIiRiVGJSNGJUYtRiVGJUYlRitGLiwkKiYsJkYlRi5GJkYlRiVGK0YuRi4=

ZW:=op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNaV0dGKCwkKiYsJiIiIiEiIiokKSwmKiQpSSJXR0YoIiIjRjFGMkYxRjEjRjFGOUYxRjFGMUY4RjJGMjcjRi4=

simplify(subs(W=1/2,%))

LCYiIiMiIiIqJCkiIiQjRiRGI0YkISIi

simplify(subs(t=1/3,Z=2-sqrt(3),Cm10overt));

LCYjIiNGIiIjIiIiKiYjRiQiIiVGJikiIiQjRiZGJUYmISIi

2*%-27;

LCQqJiMiI0YiIiMiIiIpIiIkI0YnRiZGJyEiIg==

We now take the square root. First consider the reciprocal of d0 collect(subs(sqrt((3*x-1)*(3*x-4))=sqrt(delta),factor(subs(delta=(3*x-1)*(3*x-4),rationalize(1/d0)))),delta,factor);

LCYqLCMiIiIiJSg9I0YlLCYqJiIiJ0YlSSJ4RzYiRiVGJUYlRiVGJSlGKiIiIyEiIiksJiomIiIkRiVGKkYlRiVGJUYuRi1GLilJJmRlbHRhR0YrI0YlRi1GJUYlKiYjRi1GJkYlRixGLkYl

Observe the minus sign here, explained just above c1:=-d1/4*sqrt(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCwkKiojIiIiIiIjRjEpIiIkI0Y0IiIlRjEpSSJ4R0YoRjJGMSksJiosRjRGMSwmKiYiIidGMUY4RjFGMUYxRjFGMUY3ISIiKSwmKiZGNEYxRjhGMUYxRjFGP0YyRj8pSSZkZWx0YUdGKEYwRjFGMSomRj5GMUY3Rj9GMUYwRjFGPzcjRi4=

subs(csgn(1/x)=1,series(subs(delta=(1-3*x)*(4-3*x),c1),x));

Ky9JInhHNiIsJComIiIkIiIiKUYnI0YoIiIlRighIiJGKCwkKiYjIiRWIyIjS0YoRilGKEYsIiIjLCQqJiMiJmA6JSIlWz9GKEYpRihGLEYnLCQqJiMiKCpmbk8iJk9iJ0YoRilGKEYsRissJComIyIrLEFNPzgiKDMnKVEpRihGKUYoRiwiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJw==

Version in the paper tc1pap:=-3^(5/4)*x/2*sqrt((1+6*x)/(1-3*x)*sqrt((4-3*x)/(1-3*x))+2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0YzFwYXBHRigsJCoqIyIiJCIiIyIiIilGMSNGMyIiJUYzSSJ4R0YoRjMpLCYqKCwmKiYiIidGM0Y3RjNGM0YzRjNGMywmRjNGMyomRjFGM0Y3RjMhIiJGQCkqJiwmRjZGM0Y/RkBGM0Y+RkAjRjNGMkYzRjNGMkYzRkRGM0ZANyNGLg==

subs(csgn(1/x)=1,series(subs(delta=(1-3*x)*(4-3*x),c1)-tc1pap,x));

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

Hmexpl:=-3^(7/4)*subs(x=x/3,tc1pap);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIbWV4cGxHRigsJCooIyIiKiIiIyIiIkkieEdGKEYzKSwmKigsJiomRjJGM0Y0RjNGM0YzRjNGMywmRjNGM0Y0ISIiRjspKiYsJiIiJUYzRjRGO0YzRjpGOyNGM0YyRjNGM0YyRjNGQEYzRjM3I0Yu

Let us check subs(y=0,User(3));

LCgqJilJInRHNiIiIiQiIiIpSSJ4R0YmIiIjRihGKComKUYlRitGKEYqRihGKEYlRig=

at (-1,0) N:=70: i:=1: ser:=coeff(subs(x=1/x,Cmser(N)), x,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=7..9,style=point);

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NiI=

series(Hmexpl,x,2);evalf(%);

KydJInhHNiIiIioiIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiIw==

KydJInhHNiIkIiIqIiIhIiIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiM=

at (-2,0) N:=90: i:=2: ser:=coeff(subs(x=1/x,Cmser(N)), x,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=6..8,style=point);

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NiI=

series(Hmexpl,x,3);evalf(%);

KylJInhHNiIiIioiIiIjIiRWIyIjSyIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiIk

KylJInhHNiIkIiIqIiIhIiIiJCIrKyt2JGYoISIqIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiQ=

Other points: the expression of H(x,y) Hexpl:=(Hmexpl+(2+x*y^2-3*x*y)/2*Hdexpl)/(1+x*y^2+x^2*y-3*x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIZXhwbEdGKComLCYqKCMiIioiIiMiIiJJInhHRihGNCksJiooLCYqJkYzRjRGNUY0RjRGNEY0RjQsJkY0RjRGNSEiIkY8KSomLCYiIiVGNEY1RjxGNEY7RjwjRjRGM0Y0RjRGM0Y0RkFGNEY0KixGMUY0LCgqJkY1RjQpSSJ5R0YoRjNGNEY0KigiIiRGNEY1RjRGRkY0RjxGM0Y0RjQsJkY0RjRGRkY8RjwpKiZGRkY0LCZGQEY0RkZGPEY0RkFGPCksJiooLCYqJkYzRjRGRkY0RjRGNEY0RjRGSUY8KSomRkxGNEZJRjxGQUY0RjRGM0Y8RkFGNEY0RjQsKiomKUY1RjNGNEZGRjRGNEZERjRGR0Y8RjRGNEY8NyNGLg==

Let us check a few points i:=0: j:=1: const:=coeff(coeff(convert(series(convert(series(Hexpl,x,4),polynom),y,4),polynom),x,j-i),y,j);evalf(const);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwkKiYjIiMiKSIiKSIiIikiIiQjRjMiIiNGM0YzNyNGLg==

JCIrVjlxYDwhIik=

N:=100:ser:=coeff( coeff(convert(Cser(N),polynom),x,i),y,j): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=11..const,style=point);

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NiI=

i:=-1: j:=1: const:=coeff(coeff(convert(series(convert(series(Hexpl,x,4),polynom),y,4),polynom),x,j-i),y,j);evalf(const);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwmKiYjIiNGIiIlIiIiKSIiJCNGMyIiI0YzISIiRjFGMzcjRi4=

JCIrMGQnM2AiISIp

N:=100:ser:=coeff( coeff(convert(Cser(N),polynom),x,i),y,j): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=11..const,style=point);

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

i:=0: j:=2: const:=coeff(coeff(convert(series(convert(series(Hexpl,x,4),polynom),y,4),polynom),x,j-i),y,j);evalf(const);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwmKiYjIiVcQiIkRyIiIiIpIiIkI0YzIiIjRjNGMyIiKiEiIjcjRi4=

JCIrbVFleUEhIik=

N:=100:ser:=coeff( coeff(convert(Cser(N),polynom),x,i),y,j): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=11..const,style=point);

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

i:=-1: j:=2: const:=coeff(coeff(convert(series(convert(series(Hexpl,x,4),polynom),y,4),polynom),x,j-i),y,j);evalf(const);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwmKiYjIiRWIyIiKSIiIikiIiQjRjMiIiNGMyEiIiMiJVxCIiNLRjM3I0Yu

JCIrcjFfej8hIik=

N:=100:ser:=coeff( coeff(convert(Cser(N),polynom),x,i),y,j): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=11..const,style=point);

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

A series in t^3 map(factor,subs(x=x/t,t=tV,Qx0))

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

subs(V=Vt^(1/3),%)

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

map(factor,series(subs(Vt=Vtsing,%),epsilon,2))

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

c:=subs(epsilon=1,op(5,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoLCQqLiMiIiIiIzdGMSkiIiQjRjEiIiNGMSwmKiZGNEYxSSJ4R0YoRjFGMUYxISIiRjEsJkY4RjEiIiVGOkYxLCwqJkY2RjEpKiZGN0YxRjtGMSNGNEY2RjFGMSomIiNhRjEpRjlGNEYxRjEqJiIkTiJGMSlGOUY2RjFGOiomIiNzRjFGOUYxRjEiIztGMUYxKUZAIyIiJkY2RjpGMTcjRi4=

Hx:=3^(5/2)*subs(x=x/3,c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNIeEdGKCwkKiwjIiIqIiIlIiIiLCZJInhHRihGM0YzISIiRjMsJkY1RjNGMkY2RjMsLComIiIjRjMpKiZGNEYzRjdGMyMiIiRGOkYzRjMqJkY6RjMpRjVGPkYzRjMqJiIjOkYzKUY1RjpGM0Y2KiYiI0NGM0Y1RjNGMyIjO0YzRjMpRjwjIiImRjpGNkYzNyNGLg==

(simplify(rationalize(Hx)))

KigsKiomLCYqJiIjPSIiIkkieEc2IkYoRihGJyEiIkYoKSwoKiQpRikiIiNGKEYoKiYiIiZGKEYpRihGKyIiJUYoI0YoRjBGKEYoKiZGJ0YoRi9GKEYoKiYiI2pGKEYpRihGKyIjT0YrRihGLEYrLCYqJkYzRihGKUYoRihGM0YrRis=

factor(x^2-5*x+4); factor((-18*x^2+63*x+36))

KiYsJkkieEc2IiIiIkYmISIiRiYsJkYkRiYiIiVGJ0Ym

LCQqKCIiKiIiIiwmKiYiIiNGJUkieEc2IkYlRiVGJUYlRiUsJkYpRiUiIiUhIiJGJUYt

Hxpap:=9/2+ 9*(1+2*x)/4/(1-x)*sqrt((4-x)/(1-x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIeHBhcEdGKCwmIyIiKiIiIyIiIioqI0YwIiIlRjIsJiomRjFGMkkieEdGKEYyRjJGMkYyRjIsJkY4ISIiRjJGMkY6KSomLCZGNUYyRjhGOkYyRjlGOiNGMkYxRjJGMjcjRi4=

series(Hx-Hxpap,x,10);

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Let us check Qser(5)

Ky9JInRHNiIiIiIiIiEsJkkieEdGJEYlSSJ5R0YkRiVGJSwoKiQpRigiIiNGJUYlKihGLUYlRihGJUYpRiVGJSokKUYpRi1GJUYlRi0sLCokKUYoIiIkRiVGJSooRjRGJUYsRiVGKUYlRiUqKEY0RiVGKEYlRjBGJUYlKiQpRilGNEYlRiVGLUYlRjQsMCokKUYoIiIlRiVGJSooRjxGJUYzRiVGKUYlRiUqKCIiJ0YlRixGJUYwRiVGJSooRjxGJUYoRiVGOEYlRiUqJClGKUY8RiVGJSomIiImRiVGKEYlRiUqJkZERiVGKUYlRiVGPCw0KiQpRihGREYlRiUqKEZERiVGO0YlRilGJUYlKigiIzVGJUYzRiVGMEYlRiUqKEZLRiVGLEYlRjhGJUYlKihGREYlRihGJUZCRiVGJSokKUYpRkRGJUYlKiYiIipGJUYsRiVGJSooIiM7RiVGKEYlRilGJUYlKiZGUUYlRjBGJUYlRkQ=

N:=100: i:=0: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=7..9,style=point);

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

series(Hxpap,x,1);evalf(%);

KydJInhHNiIiIioiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLQ==

KydJInhHNiIkIiIqIiIhRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLg==

N:=100: i:=1: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=11..15.5,style=point);

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Ig==

series(Hxpap,x,2);evalf(%);

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KylJInhHNiIkIiIqIiIhRickIisrK3Y9OiEiKSIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YrIiIj

N:=100: i:=2: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=14..19.5,style=point);

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

series(Hx,x,3);evalf(%);

KytJInhHNiIiIioiIiEjIiRWIyIjOyIiIiMiJS5eIiRjIyIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqIiIk

KytJInhHNiIkIiIqIiIhRickIisrK3Y9OiEiKSIiIiQiK3YkZkwqPkYqIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRisiIiQ=

The functional equation for a harmonic function in C, with steps in the original step set S Heq:=collect(-1/2*numer(H(x,y)-(Hm(x)+(2+x*y^2-3*x*y)/2*Hd(y))/(1+x*y^2+x^2*y-3*x*y)),[H,Hd,Hm],expand)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIZXFHRigsKComLCoqJilJInhHRigiIiMiIiJJInlHRihGNSEiIiomRjNGNSlGNkY0RjVGNyooIiIkRjVGM0Y1RjZGNUY1RjVGN0Y1LUkiSEdGKDYkRjNGNkY1RjUqJiwoKigjRjVGNEY1RjNGNUY5RjVGNSooI0Y7RjRGNUYzRjVGNkY1RjdGNUY1RjUtSSNIZEdGKDYjRjZGNUY1LUkjSG1HRig2I0YzRjU3I0Yu

co:=coeff(%,Hd(y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwoKigjIiIiIiIjRjFJInhHRihGMSlJInlHRihGMkYxRjEqKCMiIiRGMkYxRjNGMUY1RjEhIiJGMUYxNyNGLg==

The kernel of the companion model, at t=1/3 ker:=expand(subs(t=1/3,1-t*tSteps))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRrZXJHRigsKiomIyIiIiIiJEYxSSJ4R0YoRjEhIiIqKEYwRjFJInlHRihGNEYzRjRGNComRjBGMUY2RjFGNEYxRjE3I0Yu

The polynomial version of the kernel that appears in the paper kerpol:=-(-x^2*y-x*y^2+3*x*y-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdrZXJwb2xHRigsKiomKUkieEdGKCIiIyIiIkkieUdGKEYzRjMqJkYxRjMpRjRGMkYzRjMqKCIiJEYzRjFGM0Y0RjMhIiJGM0YzNyNGLg==

factor(kerpol/ker)

LCQqKCIiJCIiIkkieEc2IkYlSSJ5R0YnRiUhIiI=

factor(rem(co^2,kerpol,x))

LCQqKiMiIiIiIiVGJSwmSSJ5RzYiRiVGJiEiIkYlKSwmRihGJUYlRioiIiNGJSwoKiZJInhHRilGJSlGKEYtRiVGJSooIiIkRiVGMEYlRihGJUYqRiVGJUYlRio=

We recognize in the factor involving x three of the four terms of kerpol. We also expect to see its discriminant disc:=factor(discrim(factor(-kerpol),x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoKihJInlHRigiIiIsJkYvRjAiIiUhIiJGMCksJkYvRjBGMEYzIiIjRjA3I0Yu

factor(co^2-disc/4*x^2);factor(%/kerpol)

LCoqJilJInhHNiIiIiMiIiJJInlHRiZGKEYoKiZGJUYoKUYpRidGKEYoKigiIiRGKEYlRihGKUYoISIiRihGKA==

IiIi

We multiply the basic equation by a well chosen term Heqmod:=-Hd(y)^2*kerpol/x^2+ collect((Hm(x)-co*Hd(y))/x^2*Heq + Hd(y)^2*kerpol/x^2,[H,Hd,Hm],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZXFtb2RHRigsKiooKS1JI0hkR0YoNiNJInlHRigiIiMiIiIsKiomKUkieEdGKEY1RjZGNEY2RjYqJkY6RjYpRjRGNUY2RjYqKCIiJEY2RjpGNkY0RjYhIiJGNkY2RjZGOUY/Rj8qJiwmKiwjRjZGNUY2LChGO0Y2Rj1GP0Y1RjZGNkY3RjZGOUY/RjFGNkY2KihGN0Y2RjlGPy1JI0htR0YoNiNGOkY2Rj9GNi1JIkhHRig2JEY6RjRGNkY2KiwjRjYiIiVGNkY0RjYsJkY0RjZGTkY/RjYpLCZGNEY2RjZGP0Y1RjZGMEY2Rj8qJilGRkY1RjZGOUY/RjY3I0Yu

factor(Heqmod/Heq)

LCQqKCMiIiIiIiNGJSwqKigtSSNIZEc2IjYjSSJ5R0YrRiVJInhHRitGJSlGLUYmRiVGJSoqIiIkRiVGKUYlRi5GJUYtRiUhIiIqJkYmRiVGKUYlRiUqJkYmRiUtSSNIbUdGKzYjRi5GJUYyRiUpRi5GJkYyRjI=

Take the factors that contain a factor tK(1/3) hxy:=-factor((op(1,Heqmod)+op(2,Heqmod))/kerpol)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoeHlHRigsJCooIyIiIiIiI0YxLCwqKi1JI0hkR0YoNiNJInlHRihGMS1JIkhHRig2JEkieEdGKEY4RjFGPEYxKUY4RjJGMSEiIiosIiIkRjFGNUYxRjlGMUY8RjFGOEYxRjEqJkYyRjEpRjVGMkYxRjEqKEYyRjFGNUYxRjlGMUY+KihGMkYxRjlGMS1JI0htR0YoNiNGPEYxRjFGMSlGPEYyRj5GMTcjRi4=

hxy:=collect(%,[H,Hd],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoeHlHRigsJiomLCYqKiMiIiIiIiNGMywoKiZJInhHRihGMylJInlHRihGNEYzRjMqKCIiJEYzRjdGM0Y5RjMhIiJGNEYzRjMpRjdGNEY8LUkjSGRHRig2I0Y5RjNGPComLUkjSG1HRig2I0Y3RjNGPUY8RjNGMy1JIkhHRig2JEY3RjlGM0YzKiYpRj5GNEYzRj1GPEYzNyNGLg==

Let us prove that hxy is a formal power series in x and y serx:=factor(subs(Hm(0)=0,Hd(y)=H(0,y),series(hxy,x,2)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ4R0YoKydJInhHRigsJCooIyIiIiIiI0YzLUkiSEdGKDYkIiIhSSJ5R0YoRjMsKiomRjVGMylGOUY0RjNGMyooIiIkRjNGNUYzRjlGMyEiIiomRjRGMy0tJkkiREdGJTYjRjM2I0Y2RjdGM0YzKiZGNEYzLS1GRDYjSSNIbUdGKDYjRjhGM0Y/RjNGP0Y/LUkiT0dGJUZFRjg3I0Yu

We get back to the functional equation satisfied by H(x,y) factor(subs(Hm(0)=0,Hd(y)=H(0,y),series(Heq,x,2)))

KydJInhHNiIsKi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiNiNJIkhHRiQ2JCIiIUkieUdGJCEiIiooI0YuIiIjRi4tRjBGMUYuKUYzRjdGLkY0KigjIiIkRjdGLkY4Ri5GM0YuRi4tLUYpNiNJI0htR0YkNiNGMkYuRi4tSSJPR0YqRi1GNw==

D1H0ysubs:=isolate(coeff(%,x,1),D[1](H)(0,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpEMUgweXN1YnNHRigvLS0mSSJER0YlNiMiIiI2I0kiSEdGKDYkIiIhSSJ5R0YoLCgqKCNGNCIiI0Y0LUY2RjdGNClGOUY9RjQhIiIqKCMiIiRGPUY0Rj5GNEY5RjRGNC0tRjI2I0kjSG1HRig2I0Y4RjQ3I0Yu

Hence h(x,y) has no negative power of x factor(subs(%,serx))

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

One step further serx:=factor(subs(Hm(0)=0,Hd(y)=H(0,y),D1H0ysubs,series(hxy,x,3)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ4R0YoKydJInhHRigsMiooIyIiIiIiI0YzLUkiSEdGKDYkIiIhSSJ5R0YoRjMtLSZJIkRHRiU2JEYzRjM2I0Y2RjdGMyEiIiooI0YzIiIlRjMpRjVGNEYzKUY5RkNGM0YzKigjIiIkRjRGM0ZERjMpRjlGSEYzRkAqKEY1RjMpRjlGNEYzLS1GPTYjSSNIbUdGKDYjRjhGM0ZAKigjIiIqRkNGM0ZERjNGS0YzRjMqKkZIRjNGNUYzRjlGM0ZMRjNGMyokKUZMRjRGM0YzKihGMkYzLS0tSSNAQEdGJTYkRj1GNEZORlBGM0Y1RjNGM0Y4LUkiT0dGJTYjRjNGMzcjRi4=

Here is the value of h(x,y) at x=0 h0y:=convert(%,polynom)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoMHlHRigsMiooIyIiIiIiI0YxLUkiSEdGKDYkIiIhSSJ5R0YoRjEtLSZJIkRHRiU2JEYxRjE2I0Y0RjVGMSEiIiooI0YxIiIlRjEpRjNGMkYxKUY3RkFGMUYxKigjIiIkRjJGMUZCRjEpRjdGRkYxRj4qKEYzRjEpRjdGMkYxLS1GOzYjSSNIbUdGKDYjRjZGMUY+KigjIiIqRkFGMUZCRjFGSUYxRjEqKkZGRjFGM0YxRjdGMUZKRjFGMSokKUZKRjJGMUYxKihGMEYxLS0tSSNAQEdGJTYkRjtGMkZMRk5GMUYzRjFGMTcjRi4=

Once again, we get back to the functional equation for H(x,y) coeff(factor(subs(Hm(0)=0,Hd(y)=H(0,y),D1H0ysubs,series(Heq,x,3))),x,2)

LDIqJiMiIiIiIiNGJS0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkRiVGJTYjSSJIR0YuNiQiIiFJInlHRi5GJSEiIiooRiRGJS1GMUYyRiUpRjQiIiVGJUYlKigiIiRGJUY3RiUpRjRGO0YlRjUqKCMiIipGJkYlRjdGJSlGNEYmRiVGJSomLS1GKjYjSSNIbUdGLjYjRjNGJUZARiVGNSooRjtGJUZCRiVGNEYlRiUqJkY3RiVGNEYlRjUqJkYkRiUtLS1JI0BAR0YrNiRGKkYmRkRGRkYlRiU=

D111H0y:=isolate(%,(D[1,1](H))(0, y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEMTExSDB5R0YoLy0tJkkiREdGJTYkIiIiRjQ2I0kiSEdGKDYkIiIhSSJ5R0YoLDAqJi1GNkY3RjQpRjkiIiVGNEY0KigiIidGNEY8RjQpRjkiIiRGNCEiIiooIiIqRjRGPEY0KUY5IiIjRjRGNCooRkdGNC0tRjI2I0kjSG1HRig2I0Y4RjRGRkY0RkMqKEZARjRGSUY0RjlGNEY0KihGR0Y0RjxGNEY5RjRGQy0tLUkjQEBHRiU2JEYyRkdGS0ZNRjQ3I0Yu

h0y:=collect(subs(D111H0y,h0y),[H],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoMHlHRigsJiosIyIiIiIiJUYxSSJ5R0YoRjEsJkYzRjFGMiEiIkYxKSwmRjNGMUYxRjUiIiNGMSktSSJIR0YoNiQiIiFGM0Y4RjFGNSokKS0tSSJER0YlNiNJI0htR0YoNiNGPUY4RjFGMTcjRi4=

disc

KihJInlHNiIiIiIsJkYjRiUiIiUhIiJGJSksJkYjRiVGJUYoIiIjRiU=

Finally, the value of h(x,0) factor(subs(H(x,0)=Hm(x)+Hd(0),series(hxy,y,1)))

KydJInlHNiIqJiktSSNIbUdGJDYjSSJ4R0YkIiIjIiIiKUYqRishIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLEYs

Quadrant harmonic function with reverse K. steps hx0:=9/4*((1+2*x)/(1-x)*sqrt((4-x)/(1-x))+2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoeDBHRigsJioqIyIiKiIiJSIiIiwmRjNGMyomIiIjRjNJInhHRihGM0YzRjMsJkYzRjNGNyEiIkY5KSomLCZGMkYzRjdGOUYzRjhGOSNGM0Y2RjNGMyNGMUY2RjM3I0Yu

Hdexpl := 9*sqrt((2*y+1)*sqrt((4-y)/(1-y))/(1-y)-2)/((1-y)*sqrt(y*(4-y)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZGV4cGxHRigsJCoqIiIqIiIiKSwmKigsJiomIiIjRjFJInlHRihGMUYxRjFGMUYxKSomLCYiIiVGMUY4ISIiRjEsJkYxRjFGOEY9Rj0jRjFGN0YxRj5GPUYxRjdGPUY/RjFGPkY9KSomRjhGMUY7RjFGP0Y9RjE3I0Yu

Hmexpl := 9*x*sqrt((1+2*x)*sqrt((4-x)/(1-x))/(1-x)+2)*(1/2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIbWV4cGxHRigsJCooIyIiKiIiIyIiIkkieEdGKEYzKSwmKigsJkYzRjMqJkYyRjNGNEYzRjNGMywmRjNGM0Y0ISIiRjspKiYsJiIiJUYzRjRGO0YzRjpGOyNGM0YyRjNGM0YyRjNGQEYzRjM3I0Yu

series(Hmexpl/x/sqrt(hx0),x,6)

KyVJInhHNiIiIiQiIiE=

series(Hdexpl/2/sqrt(-(subs(x=y,hx0)-subs(x=0,hx0))/disc),y,6)

KydJInlHNiIiIiQiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

subs(x=1,y=1,eqC);isolate(%,Cm(1));

LCgqJiwmIiIiRiUqJiIiJEYlSSJ0RzYiRiUhIiJGJS1JIkNHRik2JEYlRiVGJUYqRiVGJSooIiIjRiVGKEYlLUkjQ21HRik2I0YlRiVGKg==

Ly1JI0NtRzYiNiMiIiIsJCooI0YnIiIjRicsJiomLCZGJ0YnKiYiIiRGJ0kidEdGJUYnISIiRictSSJDR0YlNiRGJ0YnRidGJ0YnRjJGJ0YxRjJGMg==

Value of I(x) at x=1 II1C11:=factor(subs(U(x,0)=Cm(x)/x,isolate(subs(y=x,x=1/x,eqC),Cm(x)),x=1,Ix));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJSTFDMTFHRigqKCksJiIiIiEiIiomIiIkRjFJInRHRihGMUYxIiIjRjEpLCYqJi1JIkNHRig2JEYxRjFGMUY1RjFGMUYxRjFGNkYxKUY1RjZGMjcjRi4=

Degree of C(1,1) factor(subs(x=1,Deltapx));

LCQqKiMiIiIiIiVGJSwmSSJWRzYiRiVGJSEiIkYlLCZGKEYlIiIjRiVGJSwoKiQpRihGLEYlRiVGKEYqRixGJUYlRio=

II1:=map(normal,subs(Dpx=Deltapx,x=1,Ixfinal));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJSTFHRigsKCosIiIjIiIiLCZJIlZHRihGMUYxISIiRjEsKCokKUYzRjBGMUYxRjNGMUYxRjFGMUY3RjQpSSZkZWx0YUdGKCNGMUYwRjFGNCooLChGNkYxRjNGMUYwRjRGMUYzRjQpLCoqJClGMyIiJUYxRjRGNkYxKiZGQUYxRjNGMUY0RkFGMUY6RjFGMSoqRjBGMSwmRjZGMUYxRjRGMSlGMkYwRjFGN0Y0RjQ3I0Yu

We write II for the invariant I(1), and first eliminate the square root involving V^4 res1:=numer(factor(resultant(II-subs(sqrt(Dpx)=sDpx,sqrt(delta)=sdelta,x=1,Ixfinal),subs(x=1,sDpx^2-Deltapx),sDpx))): nops(res1); degree(res1,II);

IiND

IiIj

Elimination of the other square root res2:=factor(resultant(res1,sdelta^2-1+V^3,sdelta)): nops(res2);

IiIj

op(1,res2);

KiQpSSJWRzYiIiIkIiIi

nops(op(2,res2));

IiNb

res2a:=op(2,res2): degree(res2a,II);

IiIl

Now we move from I to C11 res:=numer(factor(subs(II=II1C11,C(1,1)=C11,t=tV,res2a))):nops(%); degree(%%,C11);

IiIk

IiIp

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

KiQpLCZJIlZHNiIiIiIiIiNGJyIiJEYn

KiQpLCZJIlZHNiIiIiJGJyEiIiIiJ0Yn

IiRSIw==

PCRJJEMxMUc2IkkiVkdGJA==

The equation of C(1,1) over Q(V) algC11V:=op(3,res): degree(%,C11);

IiIp

collect(algC11V,C11,factor)

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

Finally, elimination of V algC11:=factor(resultant(algC11V,algV,V)): nops(factor(%));degree(%%,C11);

IiQoUg==

IiND

Let us check n:=20: series(subs(C11=subs(x=1,y=1,Cser(n)),algC11),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

Hence the degree is 24 Asymptotics II1;

LCgqLCIiIyIiIiwmSSJWRzYiRiVGJSEiIkYlLCgqJClGJ0YkRiVGJUYnRiVGJUYlRiVGLEYpKUkmZGVsdGFHRigjRiVGJEYlRikqKCwoRitGJUYnRiVGJEYpRiVGJ0YpKSwqKiQpRiciIiVGJUYpRitGJSomRjZGJUYnRiVGKUY2RiVGL0YlRiUqKkYkRiUsJkYrRiVGJUYpRiUpRiZGJEYlRixGKUYp

We need to study all dominant sigularities of V algV;

LCZJIlZHNiIhIiIqJkkidEdGJCIiIiwmKiQpRiMiIiRGKEYoIiIjRihGKEYo

factor(discrim(algV,V));solve(%);

LCQqKiIiJSIiIkkidEc2IkYlLCZGJSEiIiomIiIkRiVGJkYlRiVGJSwoKiYiIipGJSlGJiIiI0YlRiVGKkYlRiVGJUYlRik=

NiYiIiEjIiIiIiIkLCYjRiUiIichIiIqJiomRihGJV4jRiVGJUYlKUYmI0YlIiIjRiVGJSwmRihGKiomLCRGLEYlRiVGLkYlRio=

Asymptotics of Vt=V^3 around 1/27 (already seen) Vtsing;

LC4iIiJGIyomKSIiJCNGIyIiI0YjKUkoZXBzaWxvbkc2IkYnRiMhIiIqJiMiIiVGJkYjRipGI0YjKigjIiNOIiNhRiNGJUYjKUYqI0YmRihGI0YsKiYjIiMhKSIjIilGIylGKkYoRiNGIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2IyokKUYqIyIiJkYoRiNGIw==

Using the defining equation of V Vsing:=t*(2+Vtsing);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZWc2luZ0dGKComSSJ0R0YoIiIiLC4iIiRGMComKUYyI0YwIiIjRjApSShlcHNpbG9uR0YoRjVGMCEiIiomIyIiJUYyRjBGOEYwRjAqKCMiI04iI2FGMEY0RjApRjgjRjJGNkYwRjkqJiMiIyEpIiMiKUYwKUY4RjZGMEYwLUkiT0dGJTYjKiQpRjgjIiImRjZGMEYwRjA3I0Yu

At 1/3 (note that the meaning of epsilon is changing here) Vsing0:=series(subs(epsilon=1-27*t^3,t=1/3*(1-epsilon),Vsing),epsilon);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzBHRigsLiIiIkYvKiQpSShlcHNpbG9uR0YoI0YvIiIjRi8hIiIqJiNGLyIiJEYvRjJGL0YvKiYjIiIlIiIqRi8pRjIjRjhGNEYvRjUqJiMiIikiI0ZGLylGMkY0Ri9GLy1JIk9HRiU2IyokKUYyIyIiJkY0Ri9GLzcjRi4=

At (-1+ I sqrt(3))/6 Vsing1:=map(simplify,series(subs(epsilon=1-27*t^3,t=1/3*(-1+I*sqrt(3))/2*(1-epsilon),convert(Vsing,polynom)),epsilon,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzFHRigsLiMiIiIiIiMhIiIqJiomRi9GMF4jRjBGMEYwKSIiJEYvRjBGMCooRi9GMCwmRjBGMiomRjVGMEY2RjBGMEYwKUkoZXBzaWxvbkdGKEYvRjBGMiooI0YwIiInRjBGOUYwRjxGMEYwKigjRjEiIipGMEY5RjApRjwjRjdGMUYwRjItSSJPR0YlNiMqJClGPEYxRjBGMDcjRi4=

map(factor,series(Vsing1/(-1+I*sqrt(3))*2,epsilon))

LCwiIiJGIyokKUkoZXBzaWxvbkc2IiNGIyIiI0YjISIiKiYjRiMiIiRGI0YmRiNGIyomIyIiJSIiKkYjKUYmI0YtRilGI0YqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiQpRiZGKUYjRiM=

Vsing2:=subs(I=-I,Vsing1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzJHRigsLiMiIiIiIiMhIiIqJiwkKiZGL0YwXiNGMEYwRjBGMCkiIiRGL0YwRjIqKEYvRjAsJiomLCRGNkYwRjBGN0YwRjJGMEYyRjApSShlcHNpbG9uR0YoRi9GMEYyKigjRjAiIidGMEY6RjBGPkYwRjAqKCNGMSIiKkYwRjpGMClGPiNGOEYxRjBGMi1JIk9HRiU2IyokKUY+RjFGMEYwNyNGLg==

II1C11;

KigpLCYiIiIhIiIqJiIiJEYlSSJ0RzYiRiVGJSIiI0YlKSwmKiYtSSJDR0YqNiRGJUYlRiVGKUYlRiVGJUYlRitGJSlGKUYrRiY=

At 1/3 map(simplify,series(subs(delta=1-V^3,V=Vsing0,sqrt(II1)),epsilon,1));

LCgqKCkiIiQjRiUiIiUiIiIpLCYiIiNGKCokKUYrI0YoRitGKCEiIkYuRigpSShlcHNpbG9uRzYiI0YlIiIpRihGKCosI0YoRjRGKEYkRihGKUYvRi1GKClGMSMiIihGNEYoRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YyNiMqJClGMSMiIipGNEYoRig=

At the other dominant singularities map(simplify,series(subs(delta=1-V^3,V=Vsing1,sqrt(II1)),epsilon,1));

LCgqJCksJCooIiInIiIiLCoqKF4jRihGKCksJkYnRigqJiwkKiYiIiRGKEYrRihGKEYoKUYxI0YoIiIjRighIiJGM0YoRjJGKEYoKiYqJkY0RihGK0YoRihGMkYoRigqJEYsRihGNUYnRjVGKCksJkYoRjUqJkYrRihGMkYoRihGNEY1RjVGM0YoRigqLikqJiwqKigsJComRidGKEYrRihGKEYoRixGKEYyRihGNSomLCQqJiIjN0YoRitGKEYoRihGMkYoRjUqJkYnRihGLEYoRigiI09GKEYoRjlGNUYzRihGMkYoKUY0RjNGKCkqJClGOkYxRihGM0YoKUkoZXBzaWxvbkc2IiNGMSIiJUYoLCoqKEY3RihGLEYoRjJGKEYoKiYqJkZRRihGK0YoRihGMkYoRigqJkY0RihGLEYoRjVGRkY1RjVGNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRk82I0ZORig=

evalf(%);

LCgsJiQiK3Uhb1JWJCEiKiIiIiomJCIrJltGb0gjRiZGJ14jRidGJ0YnRicqJiwmJCIrJ2UsW0cpISM2RicqJiQiKzpTLWI3RiZGJ0YrRidGJ0YnKUkoZXBzaWxvbkc2IiMiIiQiIiVGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGNjYjRjVGJw==

-simplify(sqrt(II1C11),symbolic);

LCQqKCwmIiIiISIiKiYiIiRGJUkidEc2IkYlRiVGJSwmKiYtSSJDR0YqNiRGJUYlRiVGKUYlRiVGJUYlRiVGKUYmRiY=

const:=3^(3/4)*sqrt(2-sqrt(2))/GAMMA(5/8); evalf(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCooKSIiJCNGMCIiJSIiIiksJiIiI0YzKiQpRjYjRjNGNkYzISIiRjlGMy1JJkdBTU1BR0YlNiMjIiImIiIpRjo3I0Yu

JCIrXiIpPjs3ISIq

Let us check. n:=100: ser:=subs(x=1,y=1,Cser(n)): plot([seq([1/i,coeff(ser,t,i)/3^(i)*i^(3/8)],i=1..n)],a=0..0.1,b=1.18..1.23,style=point);

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=