Multivariable Al-Salam & Carlitz polynomials associated with the type A q-Dunkl kernel

1997/06/05 by T. H. Baker, Baker, T. H., P. J. Forrester +1 · 3 citations
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #q-alg

paper · pdf · doi:10.48550/arxiv.q-alg/9706006

LaTeX 2.09, 25 pages; the relationship with big q-Jacobi polynomials clarified

arxiv created 1997/06/12 · arxiv updated 2009/11/30

Abstract

The Al-Salam & Carlitz polynomials are q-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q-analogue of the type-A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a q-analogue of the Mehta integral, and the corresponding normalization of the multivariable Al-Salam & Carlitz polynomials is derived from a Pieri-type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big q-Jacobi polynomials.

Cited by

• Superintegrability for some (q,t)-deformed matrix models
• Spectral analysis of q-deformed unitary ensembles with the Al-Salam--Carlitz weight
• On the superintegrability of the Gaussian β ensemble and its (q,t) generalisation

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