Meixner polynomials in several variables satisfying bispectral difference equations

TL;DR

This paper constructs a parametric family of multivariable Meixner polynomials and demonstrates their bispectrality through two commuting difference operator sets linked by an involution.

Contribution

It introduces a new framework for multivariable Meixner polynomials with a bispectral involution and associated commuting difference operators.

Findings

Defined a parameter set for multivariable Meixner polynomials.

Established bispectral involution connecting variables and degrees.

Constructed two sets of commuting difference operators diagonalized by the polynomials.

Abstract

We construct a set $M_d$ whose points parametrize families of Meixner polynomials in $d$ variables. There is a natural bispectral involution $b$ on $M_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of $d$ commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution $b$.