  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key conjecture in an unpublished manuscript of Macdonald (2013).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.
