In this review, we give an overview of several recent generalizations of the Fourier transform,
related to either the Lie algebra or the Lie superalgebra. In the former case, one obtains …

We study the two-parameter family of unitary operators F k, a= exp (i π 2 a (2〈 k〉+ d+ a− 2))
exp (i π 2 a Δ k, a), which are called (k, a)-generalized Fourier transforms and defined by the …

We prove that the spherical mean value of the Dunkl-type generalized translation operator
τ^ y τ y is a positive L^ p L p-bounded generalized translation operator T^ t T t. As …

The aims of these lecture notes are twofold. On the one hand, they are meant as an
introduction to rational and trigonometric Dunkl theory, starting with the historical examples …

Abstract The Dunkl–Coulomb system in three-dimensions is introduced. The energy
spectrum and the wave functions of the system are solved by means of spectrum generating …

In this paper, we consider the normalized Bessel function of index α>− 1 2, we find an
integral representation of the term xnj α+ n (x) j α (y). This allows us to establish a product …

In this paper we study a translation operator associated with the n-dimensional (k, 1)-
generalized Fourier transform, where k is a multiplicity function for the Dunkl operators. In …

In this paper, a family of radial deformations of the realization of the Lie superalgebra
$\mathfrak {osp}(1| 2) $ in the theory of Dunkl operators is obtained. This leads to a Dirac …

In this paper, a new method is developed to obtain explicit and integral expressions for the
kernel of the (κ, a)-generalized Fourier transform for κ= 0. In the case of dihedral groups, this …

Abstract The Dunkl–Coulomb system in the plane is considered. The model is defined in
terms of the Dunkl Laplacian, which involves reflection operators, with an r− 1 potential. The …