Combinatorics of $(q,y)$-Laguerre polynomials and their moments

Abstract

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre polynomials is given using a colored version of Foata-Strehl's Laguerre configurations with suitable statistics. When $\alpha\geq 0$, the corresponding moments are described using certain classical statistics on permutations, and the linearization coefficients are proved to be a polynomial in $y$ and $q$ with nonnegative integral coefficients.