
SIGMA 6 (2010), 090, 12 pages arXiv:1007.4327
http://dx.doi.org/10.3842/SIGMA.2010.090
On a Family of 2Variable 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 2variable
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 5term recurrence relation
satisfied by these polynomials.
Key words:
hypergeometric functions; Krawtchouk polynomials in 1 and 2 variables; AppellKampede Feriet functions; integral representations; transition probability kernels; recurrence relations.
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