  Orthogonal polynomials for the multivariate hypergeometric distribution are
defined on lattices in polyhedral domains in $\RR^d$. 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 relation between the index set of orthogonal polynomials and the
lattice set in 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.
