it's related to an invariant in fact I would like to solve this :
Let $x,y$ be real positive numbers such that : $$x^n-y^n=1$$
My question is :
What's the functions such that :
$$g(z)^n+h(z)^n=1$$
It's well-know that we have :
$$ch(x)^2-sh(x)^2=1$$
And a little bit more curious we have :
$$[T_n(\frac{x+x^{-1}}{2})]^2-(\frac{x+x^{-1}}{2})^2[U_{n-1}(\frac{x+x^{-1}}{2})]^2=1$$
With $T_n$ is the polynomials (of the first kind) of Tchebytchev and $U_n$ the second .
But I have no idea for the other cases . Maybe the polynomials of Tchebytchev are the solution ...
Thanks in advance