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

SIGMA 6 (2010), 090, 12 pages      arXiv:1007.4327      https://doi.org/10.3842/SIGMA.2010.090

On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

F. Alberto Grünbaum a and Mizan Rahman b
a) Department of Mathematics, University of California, Berkeley, CA 94720, USA
b) Department of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6

Received July 25, 2010, in final form December 01, 2010; Published online December 07, 2010

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.

Key words: hypergeometric functions; Krawtchouk polynomials in 1 and 2 variables; Appell-Kampe-de Feriet functions; integral representations; transition probability kernels; recurrence relations.

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