Crossings, Motzkin paths and moments

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain qq-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulae for the moments of the qq-Laguerre and the qq-Charlier polynomials, in the style of the Touchard–Riordan formula (which gives the moments of some qq-Hermite polynomials, and also the distribution of crossings in matchings).Our method mainly consists of the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular, we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard–Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.