On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

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.

References in corpus (6)

• Bispectrality of multivariable Racah-Wilson polynomials
• Discrete orthogonal polynomials and difference equations of several variables
• A Probablistic Origin for a New Class of Bivariate Polynomials
• The Rahman polynomials and the Lie algebra sl_3(C)
• The Rahman Polynomials Are Bispectral
• Orthogonality Relations for Multivariate Krawtchouk Polynomials

Cited by in corpus (6)

• The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
• A Lie theoretic interpretation of multivariate hypergeometric polynomials
• Orthogonality Relations for Multivariate Krawtchouk Polynomials
• A superintegrable finite oscillator in two dimensions with SU(2) symmetry
• Meixner polynomials in several variables satisfying bispectral difference equations
• Quantum Walks on Graphs of the Ordered Hamming Scheme and Spin Networks