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Journal of Lie Theory 26 (2016), No. 2, 439--477
Copyright Heldermann Verlag 2016

Multivariate Meixner, Charlier and Krawtchouk Polynomials According to Analysis on Symmetric Cones

Genki Shibukawa
Dept. of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

We introduce some multivariate analogues of Meixner, Charlier and Krawtchouk polynomials, and establish their main properties by using analysis on symmetric cones, that is, duality, degenerate limits, generating functions, orthogonality relations, difference equations, recurrence formulas and determinant expressions. A particularly important and interesting result is that "the generating function of the generating functions" for the Meixner polynomials coincides with the generating function of the Laguerre polynomials, which has previously not been known even for the one variable case. Actually, main properties for the multivariate Meixner, Charlier and Krawtchouk polynomials are derived from some properties of the multivariate Laguerre polynomials by using this key result.

Keywords: Multivariate analysis, discrete orthogonal polynomials, symmetric cones, spherical polynomials, generalized binomial coefficients.

MSC: 32M15, 33C45, 43A90

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