We introduce and study the Dunkl symmetric systems. We prove the wellposedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.

An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [Ó. Ciaurri and J. L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007), 2939–2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of Uq(gln). In the GelfandTsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald’s operators to a simple kernel. An essential ingredient in the proof is ...

The structure of orthogonal polynomials on R with the weight function |x1 − x2| |x1x2|1e 2 1+x 2 2)/2 is based on the Dunkl operators of type B2. This refers to the full symmetry group of the square, generated by reflections in the lines x1 = 0 and x1− x2 = 0. The weight function is integrable if k0, k1, k0 +k1 > − 1 2 . Dunkl operators can be defined for polynomials taking values in a module o...

A new family of AN -type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin H...

In this paper, we introduce the notion of Dunkl-classical orthogonal polynomials. Then, we show that generalized Hermite and generalized Gegenbauer polynomials are the only Dunkl-classical symmetric orthogonal polynomials by solving a suitable differential-difference equation. 2006 Elsevier Inc. All rights reserved.

In this paper, using a generalized Jacobi-Dunkl translation operator, we prove a generalization of Titchmarsh’s theorem for functions in the k-JacobiDunkl-Lipschitz class defined by the finite differences of order k ∈ N∗ and Sobolev spaces associated with the Jacobi-Dunkl operator.

For a better understanding of the physical systems, even at quantum level, thermal quantities can be investigated. Recently, we realized that parity system also examined simultaneously, by substituting Dunkl operator with ordinary differential in mechanics. In this manuscript, consider two relativistic Dunkl-oscillators and investigate their well-known statistical methods. Besides, establish re...

The aim of this paper is to prove new uncertainty principles for an integral operator T with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function f ∈ L(R, μ) is highly localized near a single point then T (f) cannot be concentrated in a set of finite measure. The second...