Hahn polynomials for hypergeometric distribution

TL;DR

This paper investigates the structure and properties of Hahn polynomials associated with the multivariate hypergeometric distribution, focusing on their orthogonality, factorization, and spectral characteristics within polyhedral domains.

Contribution

It provides a detailed analysis of classical Hahn polynomials with negative parameters, revealing their factorization and relation to lattice polyhedra in multivariate settings.

Findings

Factorization formulas for Hahn polynomials are derived.

Connections between polynomial index sets and lattice domains are established.

Multivariate Hahn polynomials exhibit bispectral properties and vanish on specific lattice polyhedra.

Abstract

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.