An algebraic interpretation of the $q$-Meixner polynomials

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.

19 pages Added AMS classification numbers, a few references and thanks for useful comments on 1st version

References in corpus (2)

• The multivariate Meixner polynomials as matrix elements of $SO(d,1)$ representations on oscillator states
• $q$-Rotations and Krawtchouk polynomials