Multivariate Hahn polynomials and difference equations

TL;DR

This work constructs a complete multivariate Hahn polynomial system as eigenfunctions of a symmetric family of difference operators on a finite lattice, with orthogonality w.r.t. the hypergeometric multinomial distribution. The approach centers on a Karlin–McGregor–type main operator $ ilde{oldsymbol{H}}_T$ and commuting minors $ ilde{oldsymbol{H}}_i$, which, after a similarity transform by the weight $oldsymbol{ ho}=ig(W(m{x};m{a},b,N)ig)^{1/2}$, yield real symmetric matrices and a separable eigenbasis consisting of type-one (single-variable) and type-two (two-variable) polynomials. The principal result is an explicit complete basis $oldsymbol{P}_{m m}(m{x};m{a},b,N)$ with a product structure and uniform total-degree eigenvalues, together with forward/backward recursions; the framework also encompasses limiting families—the multivariate Krawtchouk and Meixner polynomials—obtained via parameter limits. By embedding birth–death process perspectives and connections to existing multivariate families, the paper broadens the domain of hypergeometric discrete orthogonal polynomials and provides concrete, analyzable eigenstructures in multiple variables.

Abstract

The multivariate Hahn polynomials are constructed explicitly as the common eigenvectors of a family of second order difference operators. They are orthogonal with respect to the hypergeometric multinomial distribution. The main difference operator is adopted from the work of Karlin-McGregor in 1975. The minor ones are the subsets of the main one containing less and less variables. These operators commute with each other. In contrast to the multivariate Krawtchouk and Rahman like polynomials derived recently, the entire multivariate Hahn polynomials are rational functions of the system parameters. Complete sets of multivariate Krawtchouk and Meixner polynomials are derived by limiting procedures.