The case q=1 (m=3,k=1) (Section 11.1)

Chebyshev polynomials

>    T:=proc(m) subs(cos(x)=x,expand(cos(m*x))): end:

>    T(3);

4*x^3-3*x

>    m:=3: k:=1: 2+2*cos(2*k*Pi/m);

1

Introduce the series K. The second invariant is

>    I2:=normal(subs(C=CC,II=I1,Q(0,y)=K(y)/t^2/y/q,q=1,sqrt(C)^m*subs(x=xsubs,T(m))));

I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...
I2 := -1/2*(-6*beta*y^5*K(y)*t+24*beta*y^4*K(y)*t^2+3*beta^3*y^4*K(y)*t^2+30*beta^3*y^5*K(y)*t+21*beta^2*y^4*K(y)*t^2-3*beta^2*y^5*K(y)*t-36*K(y)*y^3*beta^2*t^3+18*K(y)^2*y^4*beta^2*t^2-18*K(y)^2*y^5*b...

>    indets(I2);

{t, y, beta, K(y)}

>    factor(series(normal(I2),y,1));

series((-t^6*(1+beta)^3)*y^(-6)+O(y^(-5)),y,-5)

The invariant equation

>    eqinv:=subs(q=1,Q(0,y)=K(y)/t^2/y/q,q=1,beta=nu-1,I2-add(c[i]*(t*I1)^i,i=0..3));

eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...
eqinv := -1/2*(-36*K(y)*y^3*(nu-1)*t^3-18*K(y)^2*y^5*(nu-1)*t+6*(nu-1)^3*K(y)^2*y^4*t^2-6*(nu-1)^3*K(y)^2*y^5*t+6*(nu-1)^3*K(y)*y^2*t^4-9*K(y)*y^6*(nu-1)*t^3-18*K(y)*y^6*(nu-1)^2*t^3-9*(nu-1)^3*K(y)*y^...

>   

Determination of the series c_i

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