A high order $q$-difference equation for $q$-Hahn multiple orthogonal polynomials

TL;DR

This paper derives a high order linear q-difference equation with polynomial coefficients for q-Hahn multiple orthogonal polynomials, explores its properties, and connects it to classical Hahn polynomials as q approaches 1.

Contribution

It introduces a new high order q-difference equation for q-Hahn multiple orthogonal polynomials and clarifies its relation to existing Hahn polynomial equations in the limit q→1.

Findings

Derived a high order q-difference equation with polynomial coefficients.

Connected the q-difference equation to classical Hahn polynomials as q→1.

Corrected and extended previous results on Hahn multiple orthogonal polynomials.

Abstract

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case.