Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 089, 18 pages      arXiv:0812.3879      https://doi.org/10.3842/SIGMA.2008.089
Contribution to the Special Issue on Dunkl Operators and Related Topics

A Probablistic Origin for a New Class of Bivariate Polynomials

Michael R. Hoare and Mizan Rahman
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada

Received September 15, 2008, in final form December 15, 2008; Published online December 19, 2008

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.

Key words: cumulative Bernoulli trials; multivariate Markov chains; 9–j symbols; transition kernel; Askey-Wilson polynomials; eigenvalue problem; trinomial distribution; Krawtchouk polynomials.

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