1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.