Multivariate Meixner, Charlier and Krawtchouk polynomials

TL;DR

This paper introduces multivariate versions of Meixner, Charlier, and Krawtchouk polynomials, establishing their key properties and revealing a novel connection between their generating functions and those of Laguerre polynomials.

Contribution

It develops new multivariate analogues of classical orthogonal polynomials and uncovers a unique generating function relationship with Laguerre polynomials.

Findings

Derived orthogonality relations and difference equations for the polynomials.

Established a novel link between the generating functions of these polynomials and Laguerre polynomials.

Provided determinant expressions and recurrence formulas for the multivariate polynomials.

Abstract

We introduce some multivariate analogues of Meixner, Charlier and Krawtchouk polynomials, and establish their main properties, that is, duality, degenerate limits, generating functions, orthogonality relations, difference equations, recurrence formulas and determinant expressions. A particularly important and interesting result is that "the generating function of the generating function" for the Meixner polynomials coincides with the generating function of the Laguerre polynomials, which has previously not been known even for the one variable case. Actually, main properties for the multivariate Meixner, Charlier and Krawtchouk polynomials are derived from some properties of the multivariate Laguerre polynomials by using this key result.