An algebraic interpretation of the $q$-Meixner polynomials

TL;DR

This paper provides an algebraic framework for understanding the $q$-Meixner polynomials using representations of quantum algebra, revealing their properties through matrix elements of $q$-pseudorotation operators.

Contribution

It introduces a novel algebraic interpretation of $q$-Meixner polynomials via $ ext{U}_q( ext{su}(1,1))$ representations and $q$-oscillator states, systematically deriving their properties.

Findings

Orthogonality relations derived within the framework

Recurrence relations established algebraically

Difference equations obtained systematically

Abstract

An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary $q$-pseudorotation operators. These operators are built from $q$-exponentials of the $\mathcal{U}_q(\mathfrak{su}(1,1))$ generators. The orthogonality, recurrence relation, difference equation, and other properties of the $q$-Mexiner polynomials are systematically obtained in the proposed framework.