Is there a direct formula that relates the weight function $w(x)$ and the generating function $g(x,z)$ of a given set of orthogonal polynomials?
From their very definitions, I know that: $$\begin{align} \int w(x)[g(x,z)]^2dx&=\int w(x) \left[\sum_n P_n(x)z^n\right]\left[\sum_m P_m(x)z^m\right]dx\\&=\sum_n\sum_mz^{n+m}\int w(x)P_n(x)P_m(x)dx\\&=\sum_n\sum_mz^{n+m}\delta_{nm}=\sum_nz^{2n}\end{align}$$
So I guess that knowing one of them will require to solve this integral equation in order to get the other.
I know that the polynomials can be obtained from the weight function by Gram-Schmidt orthogonalization, and from the generating function by extracting the moments. However I would like to know if there is a more concise relationship between them.
Thanks for the help!