On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

TL;DR

This paper provides a hypergeometric proof of the orthogonality and recurrence relations for a family of 2-variable Krawtchouk polynomials, linking them to quantum angular momentum and probability models.

Contribution

It offers a new hypergeometric proof of orthogonality conditions and derives a 5-term recurrence relation for these polynomials, clarifying their geometric significance.

Findings

Orthogonality conditions established with geometric interpretation

Derived a 5-term recurrence relation for the polynomials

Linked polynomials to quantum angular momentum and probability models

Abstract

We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman [SIGMA 4 (2008), 089, 18 pages] as a limit of the 9-j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a "poker dice" type probability model. The proof in this paper derives and makes use of the necessary and sufficient conditions of orthogonality in establishing orthogonality as well as indicating their geometrical significance. We also derive a 5-term recurrence relation satisfied by these polynomials.