Multivariable Al-Salam & Carlitz polynomials associated with the type A q-Dunkl kernel
Abstract
The Al-Salam & Carlitz polynomials are -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 -analogue of the type- 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 -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 -Jacobi polynomials.
Cite
@article{arxiv.q-alg/9706006,
title = {Multivariable Al-Salam & Carlitz polynomials associated with the type A q-Dunkl kernel},
author = {T. H. Baker and P. J. Forrester},
journal= {arXiv preprint arXiv:q-alg/9706006},
year = {2008}
}
Comments
LaTeX 2.09, 25 pages; the relationship with big q-Jacobi polynomials clarified