  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.
