Some minor calculations in Section 11 (Lemmas 19 and 20)

End of the proof of Lemma 19

>    x1:=((nu-1)*(4-q)*z+q*nu*(z^2-z)+(nu-1)*(q-2))/2/sqrt(q*nu^2*(z^2-z)^2+(nu-1)*(4*(nu-1)+q)*(z^2-z)+(nu-1)^2);

x1 := 1/2*((nu-1)*(-q+4)*z+q*nu*(z^2-z)+(-2+q)*(nu-1))/(nu^2*q*z^4-2*nu^2*q*z^3+nu^2*q*z^2-8*nu*z^2+8*z*nu-q*z^2+q*z+4*z^2-4*z+q*nu*z^2-nu*z*q+4*nu^2*z^2-4*z*nu^2+nu^2+1-2*nu)^(1/2)

>    x2:=subs(z=1-z,x1);

x2 := 1/2*((nu-1)*(-q+4)*(1-z)+q*nu*((1-z)^2-1+z)+(-2+q)*(nu-1))/(nu^2*q*(1-z)^4-2*nu^2*q*(1-z)^3+nu^2*q*(1-z)^2-8*nu*(1-z)^2+8*(1-z)*nu-q*(1-z)^2+q*(1-z)+4*(1-z)^2-3+4*z+q*nu*(1-z)^2-nu*(1-z)*q+4*nu^2...
x2 := 1/2*((nu-1)*(-q+4)*(1-z)+q*nu*((1-z)^2-1+z)+(-2+q)*(nu-1))/(nu^2*q*(1-z)^4-2*nu^2*q*(1-z)^3+nu^2*q*(1-z)^2-8*nu*(1-z)^2+8*(1-z)*nu-q*(1-z)^2+q*(1-z)+4*(1-z)^2-3+4*z+q*nu*(1-z)^2-nu*(1-z)*q+4*nu^2...

>    factor(x1^2+x2^2-(q-2)*x1*x2-q*(4-q)/4);

0

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End of the proof of Lemma 20

>    unassign('m'):

The value of LHS, when  t=0  (Eq. 82). We denote z=t/y

>    Jinv:=subs(y=t/z,C=CC,II=(K-z+z^2)/t,t=0,CC^(m/2)*TT(xsubs));

Jinv := (q*(1+beta)^2*(K-z+z^2)^2+beta*(4*beta+q)*(K-z+z^2)+beta^2)^(1/2*m)*TT(1/2*(-q+q*(1+beta)+q*(1+beta)*(K-z+z^2)-2*beta)/(q*(1+beta)^2*(K-z+z^2)^2+beta*(4*beta+q)*(K-z+z^2)+beta^2)^(1/2)-1/2*(q-4...
Jinv := (q*(1+beta)^2*(K-z+z^2)^2+beta*(4*beta+q)*(K-z+z^2)+beta^2)^(1/2*m)*TT(1/2*(-q+q*(1+beta)+q*(1+beta)*(K-z+z^2)-2*beta)/(q*(1+beta)^2*(K-z+z^2)^2+beta*(4*beta+q)*(K-z+z^2)+beta^2)^(1/2)-1/2*(q-4...

The coefficient of  K*z , ie [x ] L_1(x,0)

>    e1:=collect(simplify(collect(factor(subs(K=0,z=0,diff(diff(Jinv,K),z))),TT,factor),symbolic),TT,factor);

e1 := -1/4*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)+1/8*beta^(m-2)*(q-4)*(-3*q^2+2*m*q^2-10*q*beta+10*m*beta*q+4*q*beta^2+2*q+8*m*beta^2-16*beta...
e1 := -1/4*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)+1/8*beta^(m-2)*(q-4)*(-3*q^2+2*m*q^2-10*q*beta+10*m*beta*q+4*q*beta^2+2*q+8*m*beta^2-16*beta...
e1 := -1/4*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)+1/8*beta^(m-2)*(q-4)*(-3*q^2+2*m*q^2-10*q*beta+10*m*beta*q+4*q*beta^2+2*q+8*m*beta^2-16*beta...

The first derivative D(TT) is zero

>    e1:=subs(D(TT)(1/2*q-1)=0,collect(simplify(collect(factor(subs(K=0,z=0,diff(diff(Jinv,K),z))),TT,factor),symbolic),TT,factor));

e1 := -1/4*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)-1/16*beta^(m-2)*(q-4)^2*(q+2*beta)*q*`@@`(D,2)(TT)(-1+1/2*q)

>    expand(subs(z=(1-sqrt(1+4*u))/2,z^2-z));

u

>    simplify(normal(series(subs(K=0,z=(1-sqrt(1+4*u))/2,Jinv),u,3)),symbolic);

series(beta^m*TT(-1+1/2*q)+1/4*beta^(m-1)*(-D(TT)(-1+1/2*q)*q^2+4*D(TT)(-1+1/2*q)*q+8*m*TT(-1+1/2*q)*beta+2*m*TT(-1+1/2*q)*q)*u+1/32*beta^(m-2)*(16*D(TT)(-1+1/2*q)*q-64*D(TT)(-1+1/2*q)*q*beta+32*D(TT)(...
series(beta^m*TT(-1+1/2*q)+1/4*beta^(m-1)*(-D(TT)(-1+1/2*q)*q^2+4*D(TT)(-1+1/2*q)*q+8*m*TT(-1+1/2*q)*beta+2*m*TT(-1+1/2*q)*q)*u+1/32*beta^(m-2)*(16*D(TT)(-1+1/2*q)*q-64*D(TT)(-1+1/2*q)*q*beta+32*D(TT)(...
series(beta^m*TT(-1+1/2*q)+1/4*beta^(m-1)*(-D(TT)(-1+1/2*q)*q^2+4*D(TT)(-1+1/2*q)*q+8*m*TT(-1+1/2*q)*beta+2*m*TT(-1+1/2*q)*q)*u+1/32*beta^(m-2)*(16*D(TT)(-1+1/2*q)*q-64*D(TT)(-1+1/2*q)*q*beta+32*D(TT)(...
series(beta^m*TT(-1+1/2*q)+1/4*beta^(m-1)*(-D(TT)(-1+1/2*q)*q^2+4*D(TT)(-1+1/2*q)*q+8*m*TT(-1+1/2*q)*beta+2*m*TT(-1+1/2*q)*q)*u+1/32*beta^(m-2)*(16*D(TT)(-1+1/2*q)*q-64*D(TT)(-1+1/2*q)*q*beta+32*D(TT)(...
series(beta^m*TT(-1+1/2*q)+1/4*beta^(m-1)*(-D(TT)(-1+1/2*q)*q^2+4*D(TT)(-1+1/2*q)*q+8*m*TT(-1+1/2*q)*beta+2*m*TT(-1+1/2*q)*q)*u+1/32*beta^(m-2)*(16*D(TT)(-1+1/2*q)*q-64*D(TT)(-1+1/2*q)*q*beta+32*D(TT)(...

>    e2:=collect(coeff(%,u,2),TT,factor);

e2 := 1/8*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)-1/16*beta^(m-2)*q*(q-4)*(-3*q+2*m*q+8*m*beta+2+4*beta^2-8*beta)*D(TT)(-1+1/2*q)+1/32*beta^(m-...
e2 := 1/8*beta^(m-2)*m*(-32*beta^2+16*m*beta^2+m*q^2+8*m*beta*q-2*q^2+4*q-8*q*beta+4*q*beta^2)*TT(-1+1/2*q)-1/16*beta^(m-2)*q*(q-4)*(-3*q+2*m*q+8*m*beta+2+4*beta^2-8*beta)*D(TT)(-1+1/2*q)+1/32*beta^(m-...

>    collect(e1+2*e2,TT,factor);

-1/8*beta^(m-2)*q*(q-4)*(-3*q+2*m*q+8*m*beta+2+4*beta^2-8*beta)*D(TT)(-1+1/2*q)-1/8*beta^(m-2)*q*beta*(q-4)^2*`@@`(D,2)(TT)(-1+1/2*q)

>    res:=subs(D(TT)(1/2*q-1)=0,%);

res := -1/8*beta^(m-2)*q*beta*(q-4)^2*`@@`(D,2)(TT)(-1+1/2*q)

Derivatives of the Chebyshev polynomials

>    diff(TT(cos(t))-cos(m*t),t);

-D(TT)(cos(t))*sin(t)+sin(m*t)*m

Donc T'(cos(2k pi/m))=0

>    diff(%,t);

`@@`(D,2)(TT)(cos(t))*sin(t)^2-D(TT)(cos(t))*cos(t)+cos(m*t)*m^2

>    subs(D(TT)(cos(t))=0,cos(m*t)=1,%);

`@@`(D,2)(TT)(cos(t))*sin(t)^2+m^2

Second derivative

>    factor(subs(`@@`(D,2)(TT)(1/2*q-1)=-m^2/(1-((q-2)/2)^2), res));

-1/2*m^2*(q-4)*beta^(m-2)*beta

>    factor(-m^2/(1-((q-2)/2)^2));

4*m^2/q/(q-4)

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