Lancaster distributions and Markov chains with multivariate Poisson–Charlier, Meixner and Hermite–Chebycheff polynomial eigenfunctions

Abstract

This paper studies new Lancaster characterizations of bivariate multivariate Poisson, negative binomial and normal distributions which have diagonal expansions in multivariate orthogonal polynomials. The characterizations extend classical Lancaster characterizations of bivariate 1-dimensional distributions. Multivariate Poisson–Charlier, Meixner and Hermite–Chebycheff orthogonal polynomials, used in the characterizations, are constructed from classical 1-dimensional orthogonal polynomials and multivariate Krawtchouk polynomials. New classes of transition functions of discrete and continuous time Markov chains with these polynomials as eigenfunctions are characterized. The characterizations obtained belong to a class of mixtures of multi-type birth and death processes with fixed multivariate Poisson or multivariate negative binomial stationary distributions.