The Dunkl-Laplace transform and Macdonald's hypergeometric series

TL;DR

This paper extends classical analysis results to the Dunkl setting for type A root systems, establishing a Dunkl-Laplace transform identity for hypergeometric functions and related kernels, with applications to hypergeometric series and inversion formulas.

Contribution

It proves a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A, generalizing classical results and connecting to Macdonald's conjectures.

Findings

Established Dunkl-Laplace transform identities for hypergeometric functions

Derived Laplace transform relations between hypergeometric series

Provided a Post-Widder inversion formula for the Dunkl-Laplace transform

Abstract

We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A and more generally, for the associated Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture in an unpublished manuscript of Macdonald (2013). Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of Knop and Sahi. Moreover, we use these results to establish Laplace transform identities between hypergeometric series in terms of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula for the Dunkl-Laplace transform.