
SIGMA 7 (2011), 119, 17 pages arXiv:1106.1835
http://dx.doi.org/10.3842/SIGMA.2011.119
A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
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, ONT, Canada, K1S 5B6
Received June 10, 2011, in final form December 19, 2011; Published online December 27, 2011
Abstract
The one variable Krawtchouk polynomials, a special case of the _{2}F_{1} function
did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the F_{1} Appel function shows up in the spectral analysis of the corresponding
transition kernel. Independently of any probabilistic consideration a certain
multivariable version of the GelfandAomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the twodimensional version of the HoareRahman work from a Lietheoretic point of view. P. Iliev then treats the general ndimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role
in the spectral analysis of the transition kernel that comes from pushing the work of HoareRahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above.
Key words:
multivariable Krawtchouk polynomials; GelfandAomoto hypergeometric functions; cumulative Bernoulli trial; poker dice.
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