We introduce the so-called Clifford–Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related ...

In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov–Dunkl space of functions for which the remainder in the generalized Taylor’s formula has a given order. We provide characterization of these spaces by the Dunkl convolution.

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.

We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type A, we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided dif...

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl [1] and are denoted by Λα, where α is a real parameter > −1/2. These operators are associated with the reflection group Z2 on R. The Dunkl kernel Eα is used to define the Dunkl transform α which was introduced by Dunkl in [2]. Rösler in [3] shows that the Dunkl kernels verify a product formul...

Let G be a noncompact semisimple Lie group with finite centre. $$X=G/K$$ the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections to Laplace-Beltrami operator are given by $$P_{\lambda }f =f*\Phi _{\lambda }$$ , where $$\Phi elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for }f$$ . Moreove...

The one-dimensional Dunkl operator Dk with a non-negative parameter k, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of Dk, satisfying this condition is studied. An operational calculus of Mikusiński type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value ...

in this paper, using a generalized translation operator, we prove theestimates for the generalized fourier-bessel transform in the space l2 on certainclasses of functions.

Abstract Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors this manuscript presented most general forms that depends on three Wigner parameters have a better tuning. In manuscript, we employ latter generalized in relativistic equation examine two dimensional harmonic and anharmonic oscillators solutions. We obtain Nikiforo...