Hahn polynomials for hypergeometric distribution

Abstract

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in Rd. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relationship between the index set of orthogonal polynomials and the lattice set in the polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral properties of the orthogonal polynomials are also discussed.