Multivariate Meixner, Charlier and Krawtchouk 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.