Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting

Abstract

We study analytic aspects of the Dunkl-type Hankel transform, which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel function and K-Bessel function generalizing those of a symmetric cone. Further, we take a look at zeta integrals and their distributional extensions in the Dunkl setting. These distributions are closely related to Dunkl-type Riesz distributions. Further, we study regularity properties of the zeta distributions and prove a functional equation relating zeta distributions and their Dunkl transform.