A Probablistic Origin for a New Class of Bivariate Polynomials

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.

Keywords

• cumulative Bernoulli trials
• multivariate Markov chains
• 9-j symbols
• transition kernel
• Askey-Wilson polynomials
• eigenvalue problem
• trinomial distribution
• Krawtchouk polynomials