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Multivariable Al–Salam & Carlitz Polynomials Associated with the Type A q –Dunkl Kernel
Author(s) -
Baker T.H.,
Forrester P.J.
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/(sici)1522-2616(200004)212:1<5::aid-mana5>3.0.co;2-s
Subject(s) - mathematics , wilson polynomials , hahn polynomials , koornwinder polynomials , discrete orthogonal polynomials , classical orthogonal polynomials , difference polynomials , orthogonal polynomials , macdonald polynomials , pure mathematics , jacobi polynomials , askey–wilson polynomials , gegenbauer polynomials , hypergeometric function , multivariable calculus , algebra over a field , control engineering , engineering
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.