On the $q$-Charlier Multiple Orthogonal Polynomials

TL;DR

This paper introduces $q$-Charlier multiple orthogonal polynomials, exploring their structural properties, explicit representations, and difference equations, advancing the understanding of $q$-analogues of classical orthogonal polynomials.

Contribution

It presents the first detailed study of $q$-Charlier multiple orthogonal polynomials, including their operators, formulas, and recurrence relations.

Findings

Derived raising and lowering operators.

Established Rodrigues-type formulas.

Obtained explicit hypergeometric representations.

Abstract

We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a $q$-analogue of the second of Appell's hypergeometric functions is given. A high-order linear $q$-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.