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Dunkl approach to slice regular functions

Giulio Binosi, Hendrik De Bie, Pan Lian

TL;DR

The work builds a bridge between Dunkl-Clifford analysis and slice regularity by characterizing sliceness via the kernel of the Dunkl-spherical Dirac operator and slice regularity via the kernel of the Dunkl-Cauchy-Riemann operator under $\gamma_{\kappa}=(1-m)/2$. It introduces a constructive path from holomorphic stem data to families of monogenic functions using the inverse Dunkl intertwining operator, extending Fueter-type mappings beyond traditional single-function lifts. The paper also reveals a shift mechanism for the index $\gamma_{\kappa}$ through Dunkl-Laplacian actions and connects these ideas to hyperbolic harmonic structures and CK-type extensions. Overall, it provides new tools to generate and study monogenic functions in the joint Dunkl-Clifford and slice-analytic framework, with promising directions for mean value properties and Cauchy-type formulas.

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.

Dunkl approach to slice regular functions

TL;DR

The work builds a bridge between Dunkl-Clifford analysis and slice regularity by characterizing sliceness via the kernel of the Dunkl-spherical Dirac operator and slice regularity via the kernel of the Dunkl-Cauchy-Riemann operator under . It introduces a constructive path from holomorphic stem data to families of monogenic functions using the inverse Dunkl intertwining operator, extending Fueter-type mappings beyond traditional single-function lifts. The paper also reveals a shift mechanism for the index through Dunkl-Laplacian actions and connects these ideas to hyperbolic harmonic structures and CK-type extensions. Overall, it provides new tools to generate and study monogenic functions in the joint Dunkl-Clifford and slice-analytic framework, with promising directions for mean value properties and Cauchy-type formulas.

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.
Paper Structure (9 sections, 15 theorems, 80 equations, 1 figure)

This paper contains 9 sections, 15 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dunkl operators
  4. Dunkl-Clifford analysis
  5. Slice regular functions
  6. Characterization of slice regular functions
  7. Fueter's theorem revisited
  8. New approach to generate monogenic functions by the inverse intertwining operator
  9. Concluding remarks

Key Result

Lemma 2.7

fck The Dunkl-Dirac operator on $\mathbb{R}^{m}$ in spherical coordinates is given by where $\underline{\omega}=\underline{x}/|\underline{x}|$, $r=|\underline{x}|$ and is the Dunkl-spherical Dirac operator, in which is the spherical Dirac operator and for any $f\in \mathcal{C}^{1}(\mathbb{R}^{m})$.

Figures (1)

  • Figure 1: Dunkl-monogenic, slice regular and hypermonogenic functions on $\Omega_D$ centered at the origin

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 3.1
  • ...and 34 more