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

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

1997-06-05

arxiv.org/abs/q-alg/9706006v2

Mathematics

Quantum Algebra

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

Scientific paper

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.

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