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A Lie-theoretic interpretation of multivariate hypergeometric polynomials

Part of: Difference equations Difference and functional equations Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  20 March 2012

Plamen Iliev
Plamen Iliev*
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA (email: iliev@math.gatech.edu)
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Abstract

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In 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra . Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d2+d+1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.

Keywords

multivariate Krawtchouk polynomialsthe Lie algebra 𝔰𝔩d+1partial difference operatorsthe bispectral problem

MSC classification

Secondary: 33C80: Connections with groups and algebras, and related topics 17B10: Representations, algebraic theory (weights) 39A14: Partial difference equations
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 991 - 1002
Copyright
Copyright © Foundation Compositio Mathematica 2012

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