On the Dunkl Version of Monogenic Polynomials

Abstract

In this note, Kelvin transform is introduced in the framework of Dunkl-Clifford analysis. It is shown that this transform preserves the class of Dunkl monogenic functions. As an application, we use it to generate Dunkl monogenic polynomials by a classical process due to Maxwell.

Generalised Symmetries and Bases for Dunkl Monogenics

“…The relation between the two Kelvin-type transforms (2.11) and (3.30) is The transform (3.30) was considered before, for example see [16] and [10]. One of the main results of those two papers is to prove that, for any polynomial monogenic f , also I κ D j I κ ( f ) is a polynomial monogenic.…”

“…We can construct monogenics of higher degree by acting with the generalised symmetries z j . Now, let β ∈ N d with |β | 1 = n. For the next result (see also [16,Prop 4.2]), we will use (3.16) and Xu's work on the equivalence between the projection operator of harmonics and the H β . Moreover, we also need Proposition 3.9 and the correspondence (3.32) between I κ and K κ .…”

“…This construction was translated to Dunkl harmonic analysis by Xu [15] and to Dunkl-Clifford analysis in [10,16]. Similar operators were also considered in the study of the conformal symmetries of the super Dirac operator [3] and on the radially deformed Dirac operator [4].…”

Generalised symmetries and bases for Dunkl monogenics

“…The transform (3.27) was considered before, for example see [16] and [10]. One of the main results of those two papers is to prove that, for any polynomial monogenic f , also I κ D j I κ ( f ) is a polynomial monogenic.…”

“…The generalised symmetries are related to the Maxwell representation in harmonic analysis [12, p.69], which was translated to Dunkl harmonic analysis by Xu [15] and to Dunkl-Clifford analysis in [10,16]. Similar operators were also considered in the study of the conformal symmetries of the super Dirac operator [3] and on the radially deformed Dirac operator [4].…”