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New Taylor and Laurent series of axially harmonic, Fueter regular and polyanalytic functions

Fabrizio Colombo, Antonino De Martino, Irene Sabadini

TL;DR

The paper addresses extending quaternionic power-type expansions to axially harmonic, Fueter-regular, and axially polyanalytic (order 2) function classes arising from the two factorizations of the Fueter map, Δ = \bar{D} D and Δ = D \bar{D}. It develops comprehensive Taylor and Laurent series frameworks (both $*$- and spherical variants) for these fine-structure function spaces, derives explicit representation formulas, and studies convergence regions using kernels, polynomials, and integral representations. The work connects slice hyperholomorphic theory to Fueter-regular theory via the Fueter mapping theorem, introduces and exploits kernels F_L, F_R, and Q-objects, and highlights applications to operator theory through an $F$-functional calculus. The results provide explicit constructive tools for spectral analysis on the $S$-spectrum in the quaternionic setting, enabling new analytic and functional-calculus techniques for Dirac-type operators. Overall, the paper advances the understanding of quaternionic function theories with refined spectral structures and broadens the toolkit for operator-theoretic applications in hypercomplex analysis.

Abstract

The Fueter-Sce mapping theorem stands as one of the most profound outcomes in complex and hypercomplex analysis, producing hypercomplex generalizations of holomorphic functions. In recent years, delving into the factorization of the second operator appearing in the Fueter-Sce mapping theorem has uncovered its potential to generate novel classes of functions and their respective functional calculi. The sets of functions obtained from this factorization and the associated functional calculi define the so-called {\em fine structures on the $S$-spectrum}. This paper aims to comprehensively investigate the function theories for the fine structures of Dirac type in the quaternionic framework, presenting new series expansions for axially harmonic, Fueter regular, and axially polyanalytic functions. These series expansions are highly nontrivial. In fact, when considering the hypercomplex realm, specifically the quaternionic or the Clifford setting, extending the concept of complex power series expansion is not immediate, and different Taylor and Laurent expansions appear with different sets of convergence. Additionally, our objectives include establishing the representation formulas for these function spaces; such formulas encode the fundamental properties of the functions and have numerous consequences. Finally, in the last section of this paper, we explain the applications of the fine structures in operator theory.

New Taylor and Laurent series of axially harmonic, Fueter regular and polyanalytic functions

TL;DR

The paper addresses extending quaternionic power-type expansions to axially harmonic, Fueter-regular, and axially polyanalytic (order 2) function classes arising from the two factorizations of the Fueter map, Δ = \bar{D} D and Δ = D \bar{D}. It develops comprehensive Taylor and Laurent series frameworks (both - and spherical variants) for these fine-structure function spaces, derives explicit representation formulas, and studies convergence regions using kernels, polynomials, and integral representations. The work connects slice hyperholomorphic theory to Fueter-regular theory via the Fueter mapping theorem, introduces and exploits kernels F_L, F_R, and Q-objects, and highlights applications to operator theory through an -functional calculus. The results provide explicit constructive tools for spectral analysis on the -spectrum in the quaternionic setting, enabling new analytic and functional-calculus techniques for Dirac-type operators. Overall, the paper advances the understanding of quaternionic function theories with refined spectral structures and broadens the toolkit for operator-theoretic applications in hypercomplex analysis.

Abstract

The Fueter-Sce mapping theorem stands as one of the most profound outcomes in complex and hypercomplex analysis, producing hypercomplex generalizations of holomorphic functions. In recent years, delving into the factorization of the second operator appearing in the Fueter-Sce mapping theorem has uncovered its potential to generate novel classes of functions and their respective functional calculi. The sets of functions obtained from this factorization and the associated functional calculi define the so-called {\em fine structures on the -spectrum}. This paper aims to comprehensively investigate the function theories for the fine structures of Dirac type in the quaternionic framework, presenting new series expansions for axially harmonic, Fueter regular, and axially polyanalytic functions. These series expansions are highly nontrivial. In fact, when considering the hypercomplex realm, specifically the quaternionic or the Clifford setting, extending the concept of complex power series expansion is not immediate, and different Taylor and Laurent expansions appear with different sets of convergence. Additionally, our objectives include establishing the representation formulas for these function spaces; such formulas encode the fundamental properties of the functions and have numerous consequences. Finally, in the last section of this paper, we explain the applications of the fine structures in operator theory.
Paper Structure (31 sections, 117 theorems, 613 equations)

This paper contains 31 sections, 117 theorems, 613 equations.

Table of Contents

  1. Introduction
  2. Functions of the quaternionic fine structure of Dirac type
  3. Slice hyperholomorphic functions
  4. Slice hyperholomorphic Laurent series
  5. Cauchy-Fueter regular functions
  6. Two classes of quaternionic polyanalytic functions
  7. Integral representations of the functions of the fine structure of Dirac type
  8. Relation between the $*$-Taylor expansion and the spherical expansion
  9. Representation formulas of the quaternionic fine structure of Dirac-type
  10. Harmonic case
  11. Axially polyanalytic functions of order $2$
  12. Axially regular case
  13. Taylor series: axially harmonic functions
  14. Axially harmonic series
  15. Harmonic spherical series
  16. ...and 16 more sections

Key Result

Theorem 2.6

Let $U$ be an axially symmetric slice domain in $\mathbb{H}$ and let $I\in\mathbb{S}$. A function $f:U \to \mathbb{H}$ is a left slice hyperholomorphic function (according to Definition sho) on $U$ if and only if for any quaternion $q=u+I_{\underline{q}}v \in U$ we have and the two functions depend on $u,v$ but not on $I_{\underline{q}}$. A function $f:U\to\mathbb{H}$ is a right slice hyperholom

Theorems & Definitions (295)

  • Definition 2.1
  • Definition 2.2: Slice Cauchy domain
  • Definition 2.3: Slice hyperholomorphic functions
  • Definition 2.4: Slice hyperholomorphic functions
  • Definition 2.5
  • Theorem 2.6: Representation formula
  • Remark 2.7
  • Example 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 285 more