Crossings, Motzkin paths and moments

Abstract

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-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 q-Laguerre and the q-Charlier polynomials, in the style of the TouchardRiordan formula (which gives the moments of some q-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 TouchardRiordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations. © 2011 Elsevier B.V. All rights reserved.