A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

TL;DR

This paper introduces a multivariable system of Krawtchouk polynomials and demonstrates their crucial role in analyzing the spectral properties of a multivariable Markov chain transition kernel, using novel methods.

Contribution

It develops a new approach to the spectral analysis of multivariable Markov chains using a multivariable Krawtchouk polynomial system, differing from previous Lie-theoretic methods.

Findings

Multivariable Krawtchouk polynomials are key in spectral analysis.

New methods provide alternative proofs of properties of these functions.

The work extends previous two-variable results to the general multivariable case.

Abstract

The one variable Krawtchouk polynomials, a special case of the $_2F_1$ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the $F_1$ Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general $n$-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above.