Crossings, Motzkin paths and Moments

TL;DR

This paper derives simple formulas for the moments of certain $q$-orthogonal polynomials using combinatorial models involving crossings, Motzkin paths, and advanced enumeration techniques, extending known formulas like Touchard-Riordan.

Contribution

It introduces a new combinatorial approach to compute moments of $q$-Laguerre and $q$-Charlier polynomials via weighted Motzkin paths and bijective decompositions, generalizing previous methods.

Findings

Derived formulas for moments of $q$-Laguerre and $q$-Charlier polynomials.

Established connections between moments and crossings in permutations and set partitions.

Extended combinatorial enumeration techniques to include continued fractions and hypergeometric series.

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 formulas for the moments of the $q$-Laguerre and the $q$-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some $q$-Hermite polynomials, and also the distribution of crossings in matchings). Our method mainly consists in 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.