Ju n 19 97 Multivariable Al - Salam & Carlitz polynomials associated with the type A q - Dunkl kernel

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 multi-variable 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.