The Dunkl-Laplace transform and Macdonald’s hypergeometric series. (English) Zbl 1530.33013
Summary: 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 Opdam-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 by I. G. Macdonald [“Hypergeometric functions I”, Preprint, arXiv:1309.4568v1]. Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of F. Knop and S. Sahi [Invent. Math. 128, No. 1, 9–22 (1997; Zbl 0870.05076)]. 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.
MSC:
| 33C67 | Hypergeometric functions associated with root systems |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 43A85 | Harmonic analysis on homogeneous spaces |
| 05E05 | Symmetric functions and generalizations |
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 44A10 | Laplace transform |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Citations:
Zbl 0870.05076Software:
DLMFReferences:
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