We continue a program generalizing classical results from the analysis on symmetric cones to Dunkl setting for root systems of type A A ...

A series expansion for Heckman-Opdam hypergeometric functions φλ is obtained for all λ ∈ a∗ C . As a consequence, estimates for φλ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the no...

Let R be a root system of type BC in a = Rr of general positive multiplicity. We introduce certain canonical weight function on Rr which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and ...

We conjecture exact estimates for the Weyl-invariant Opdam–Cherednik hypergeometric functions. prove root system $A_n$ and all rank 1 cases. provide other evidence that might be true in general.

In this note we give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in [O1, O2] using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use mu...

Abstract In this paper, we discuss a generalization to the Cherednik–Opdam integral operator an abstract space of Boehmians. We introduce sets Boehmians and establish delta sequences certain class convolution products. Then prove that extended is linear, bijective continuous with respect convergence generalized spaces Moreover, derive embeddings properties theory. obtain inversion formula provi...

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials. 1. Introduction In a to date unpublished but well-known manuscript, Macdonald introduced certain families of multivariable orthogonal polynomials a...

We study the rational Cherednik algebra Hc of type G(r, 1, n) by means of the Dunkl-Opdam operators introduced in [DuOp] and the generalized Jack polynomials introduced in [Gri]. Our main result is a characterization of the set of parameters c for which Hce+Hc 6= Hc, where e+ is the symmetrizing idempotent for G(r, 1, n). Such parameters are called aspherical. The proof is parallel to that in [...