The case q=1 (m=3, k=1)  (Section 10.1)

The 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

>    I2:=normal(subs(C=CC,II=I1,q=1,yinv=1/y,M(1,y)=M(y),normal(sqrt(C)^m*subs(x=xsubs,T(m))))):

>    factor(series(normal(I2),y=1,1));

series(1/2*t^3*(nu-2)*(2*nu-1)*(nu+1)*(y-1)^(-3)+O((y-1)^(-2)),y=-(-1),-2)

The invariant equation:

>    eqinv:=subs(M(1,y)=M(y),q=1,I2-add(c[i]*I1^i,i=0..3)):

>   

>    indets(eqinv);

{M(y), y, w, t, nu, c[0], c[1], c[2], c[3]}

>   

>   

Determination of the series c_i. One obtains an equation with a single catalytic variable y, and one additional unknown series K(1) (or M(1))

>   

>