Generating function for products of laguerre polynomials?

Question

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher powers.

Is there a way of reexpressing this expression which would make its evaluation easier?

Answer 1

For higher powers, we may write $L_n(x)^m=\sum_{j=0}^{nm}c(m,j)L_j(x)$ where the Fourier coefficient $$c(m,j):=\int_0^\infty L^m_n(x)L_j(x)e^{-x}dx$$ is a nonnegative integer and has a combinatorial interpretation in terms of derangements.