A Probablistic Origin for a New Class of Bivariate Polynomials

TL;DR

This paper introduces a probabilistic method to generate a new class of orthogonal polynomials in two discrete variables, linked to a bivariate Markov chain with a convolution-based transition kernel, expanding the theory of orthogonal polynomials.

Contribution

It presents a novel probabilistic framework for deriving bivariate orthogonal polynomials via an eigenvalue problem associated with a specific Markov chain.

Findings

Discovery of a new class of bivariate orthogonal polynomials.

Connection between these polynomials and a bivariate Markov chain.

Potential for extending the probabilistic approach to other polynomial classes.

Abstract

We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the discovery of an exactly soluble eigenvalue problem corresponding to a bivariate Markov chain with a transition kernel formed by a convolution of simple binomial and trinomial distributions. The solution of the relevant eigenfunction problem, giving the spectral resolution of the kernel, leads to what we believe to be a new class of orthogonal polynomials in two discrete variables. Possibilities for the extension of this approach are discussed.