Hankel transform, [formula omitted]-Bessel functions and zeta distributions in the Dunkl setting

Abstract

We study analytic properties of a Hankel transform for the type A Dunkl setting with arbitrary multiplicity parameter k≥0 which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel functions and K-Bessel functions, generalizing those of a symmetric cone which occur when the multiplicity parameter k is chosen to be twice the Peirce dimension constant of the cone. In particular, we show that our K-Bessel function is a solution of a type B system of Dunkl operators which is a generalization of the Bessel system on a symmetric cone. Furthermore, we define zeta distributions and prove a functional equation relating them with their type B Dunkl transform. In the case of symmetric cones the aforementioned functional equation reduces to the known functional equation between zeta distributions and their Fourier transform. For the proof of this functional equation, the K-Bessel function and its analytic properties turn out to be crucial. Finally, we examine which of the zeta distributions are given by a positive measure.