Biorthogonal Expansion of Non-Symmetric Jack Functions

Abstract

Abstract. We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

The Dunkl-Laplace transform and Macdonald's hypergeometric series

“…where compared to Macdonald's version, the Bessel function is replaced by the Dunkl kernel E k of type A n−1 and multiplicity k. This transform was already considered by Baker and Forrester [BF98] and later used in [SZ07], but just as for Macdonald's transform, convergence issues had remained open for a long time. Formula (1.3) generalizes a Laplace transform identity for spherical polynomials on a symmetric cone, which is in turn a consequence of the following important Laplace transform identity for the generalized power functions ∆ s , s = (s 1 , .…”

Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials

“…Also, see the papers by Davidson-Ólafsson [5] and Aristidou-Davidson-Ólafsson [2]. Further, for an arbitrary real value of the multiplicity d, the multivariate Meixner-Pollaczek polynomials are defined by Sahi-Zhang [15] in the setting of Heckman-Opdam and Cherednik-Opdam transforms, related to symmetric and non-symmetric Jack polynomials, and generating formulae for them are established. However the case where the parameter φ is involved has not been studied so far.…”