The multivariate Meixner polynomials as matrix elements of $SO(d,1)$ representations on oscillator states

Abstract

The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the $SO(d,1)$ group on oscillator states. These polynomials depend on $d$ discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2,1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk and Meixner polynomials. It is explained how these results generalize directly to $d$ variables.

22 pages

References in corpus (1)

• The Rahman Polynomials Are Bispectral

Cited by in corpus (7)

• The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states
• Spin lattices, state transfer and bivariate Krawtchouk polynomials
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• Lancaster distributions and Markov chains with Multivariate Poisson-Charlier, Meixner and Hermite-Chebycheff polynomial eigenfunctions
• An algebraic interpretation of the $q$-Meixner polynomials
• Rahman polynomials