Dunkl approach to slice regular functions

TL;DR

This paper links Dunkl analysis with slice regular functions in Clifford algebras, providing new characterizations and a method to generate monogenic functions from holomorphic ones using Dunkl operators.

Contribution

It establishes a novel connection between Dunkl operators and slice regular functions, enabling construction of monogenic functions from holomorphic functions via Dunkl analysis.

Findings

Characterization of slice functions via Dunkl-spherical Dirac operator

Characterization of slice regular functions via Dunkl-Cauchy-Riemann operator

New method to construct monogenic functions from holomorphic functions

Abstract

In this paper, we establish a connection between Dunkl analysis and slice analysis in the setting of Clifford algebras. Specifically, we show that a Clifford algebra-valued function is slice if, and only if, it belongs to the kernel of the Dunkl-spherical Dirac operator and that a slice function is slice regular if, and only if, it lies in the kernel of the Dunkl-Cauchy-Riemann operator for a suitable parameter. Based on this correspondence and the inverse Dunkl intertwining operator, we propose a new method to construct a family of classical monogenic functions from a given holomorphic function, in the spirit of Fueter theorem.