Multivariable Al-Salam & Carlitz Polynomials Associated with the TypeA q-Dunkl Kernel

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.

BCn-symmetric polynomials

“…In this limit, our difference operators converge to those of Knop and Sahi [K1, S1]. In particular, we see immediately that the shifted Macdonald polynomials are limits of BC/C-type Macdonald polynomials in multiple ways [BF2], as they are eigenfunctions of limiting versions of D (n) (u 1 , u 2 ; q, t). The action on s becomes trivial in the limit.…”

Isomorphisms of type A affine Hecke algebras and multivariable orthogonal polynomials

“…But from [1] we know that this same eigenvalue equation applies with n i=1 h i replaced by t 1−nH . The result now follows from the fact that {V (a) η + } are a basis for symmetric functions.…”

“…We remark that this generating function could also be derived in a manner similar to that done in the symmetric case [1], namely by applying the operator ( Y −1 i ) (z) to both sides of (3.21) and deducing that E…”

“…and an integral formula for the kernel 0 ψ 0 (y; z) given in [1,Prop 4.8], and consider the generating function…”

“…In [1] it was observed that the symmetric ASC polynomials V (a) λ (x) coincide (up to a factor and change of variables) with the shifted Macdonald polynomials when a = 0. We show now that this behaviour carries over to the non-symmetric case.…”