Moments of discrete classical $q$-orthogonal polynomial ensembles

TL;DR

This paper explores discrete $q$-analogues of classical orthogonal polynomial ensembles, deriving moment representations in terms of hypergeometric series and revealing connections to hypergeometric orthogonal polynomials.

Contribution

It provides new explicit formulas for moments of discrete $q$-Hermite and $q$-Laguerre ensembles using hypergeometric series, extending prior results.

Findings

Moments expressed as basic hypergeometric series

Representation of moments as hypergeometric orthogonal polynomials

Three-term recurrences in the order of moments

Abstract

We consider some discrete $q$-analogues of the classical continuous orthogonal polynomial ensembles. Building on results due to Morozov, Popolitov and Shakirov, we find representations for the moments of the discrete $q$-Hermite and discrete $q$-Laguerre ensembles in terms of basic hypergeometric series. We find that when the number of particles is suitably randomised, the moments may be represented as basic hypergeometric orthogonal polynomials, with corresponding three-term recurrences in $k$, the order of the moments.