A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

Abstract

The one variable Krawtchouk polynomials, a special case of the ${}_2{F}_1$ 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.

References in corpus (5)

• A Probablistic Origin for a New Class of Bivariate Polynomials
• A Lie theoretic interpretation of multivariate hypergeometric polynomials
• The Rahman polynomials and the Lie algebra sl_3(C)
• Orthogonality Relations for Multivariate Krawtchouk Polynomials
• Two Stochastic Models Of a Random Walk In The U(n)-Spherical Duals Of U(n + 1)

Cited by in corpus (11)

• The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
• Entanglement of inhomogeneous free fermions on hyperplane lattices
• The multivariate Meixner polynomials as matrix elements of $SO(d,1)$ representations on oscillator states
• A superintegrable finite oscillator in two dimensions with SU(2) symmetry
• Meixner polynomials in several variables satisfying bispectral difference equations
• Spin lattices, state transfer and bivariate Krawtchouk polynomials
• Markov Chains Generated by Convolutions of Orthogonality Measures
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• Lancaster distributions and Markov chains with Multivariate Poisson-Charlier, Meixner and Hermite-Chebycheff polynomial eigenfunctions
• Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice