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A Borel-Pompeiu formula in a $(q,q')$-model of quaternionic analysis

José Oscar González-Cervantes, Juan Bory-Reyes, Irene Sabadini

TL;DR

This work extends quaternionic analysis to a two-parameter $(q,q')$-calculus by introducing multi-parameter deformations of the $ψ$-Fueter operator on quaternion-valued functions in $\mathbb{R}^4$, via $(q,q')$-derivatives. It defines left and right deformed operators ${}^{ψ}\partial_{\vec{q},\{\vec{q}\}'}$ and their $r$-variants, and proves $(q,q')$-Stokes and $(q,q')$-Borel-Pompeiu formulas, yielding Cauchy-type representations with a quaternionic Cauchy kernel $K_{ψ}$. The paper also discusses reductions to the classical $ψ$-Fueter framework in the appropriate limits and lays groundwork for a multi-variable $(q,q')$-calculus in quaternionic (and Clifford) analysis. Overall, it inaugurates quaternionic results in $(q,q')$-calculus, opening avenues for quantum-deformed hypercomplex analysis with potential applications in mathematical physics.

Abstract

The study of $ψ-$hyperholomorphic functions defined on domains in $\mathbb R^4$ with values in $\mathbb H$, namely null-solutions of the $ψ-$Fueter operator, is a topic which captured great interest in quaternionic analysis. This class of functions is more general than that of Fueter regular functions. In the setting of $(q,q')-$calculus, also known as post quantum calculus, we introduce a deformation of the $ψ-$Fueter operator written in terms of suitable difference operators, which reduces to a deformed $q$ calculus when $q'=1$. We also prove the Stokes and Borel-Pompeiu formulas in this context. This work is the first investigation of results in quaternionic analysis in the setting of the $(q,q')-$calculus theory.

A Borel-Pompeiu formula in a $(q,q')$-model of quaternionic analysis

TL;DR

This work extends quaternionic analysis to a two-parameter -calculus by introducing multi-parameter deformations of the -Fueter operator on quaternion-valued functions in , via -derivatives. It defines left and right deformed operators and their -variants, and proves -Stokes and -Borel-Pompeiu formulas, yielding Cauchy-type representations with a quaternionic Cauchy kernel . The paper also discusses reductions to the classical -Fueter framework in the appropriate limits and lays groundwork for a multi-variable -calculus in quaternionic (and Clifford) analysis. Overall, it inaugurates quaternionic results in -calculus, opening avenues for quantum-deformed hypercomplex analysis with potential applications in mathematical physics.

Abstract

The study of hyperholomorphic functions defined on domains in with values in , namely null-solutions of the Fueter operator, is a topic which captured great interest in quaternionic analysis. This class of functions is more general than that of Fueter regular functions. In the setting of calculus, also known as post quantum calculus, we introduce a deformation of the Fueter operator written in terms of suitable difference operators, which reduces to a deformed calculus when . We also prove the Stokes and Borel-Pompeiu formulas in this context. This work is the first investigation of results in quaternionic analysis in the setting of the calculus theory.
Paper Structure (7 sections, 9 theorems, 99 equations)

This paper contains 7 sections, 9 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternionic analysis
  4. Basics on the $(q,q')-$calculus
  5. On a quaternionic $(q,q')-$derivative
  6. The $(q,q')-$deformed quaternionic analysis
  7. Main Results

Key Result

Proposition 2.6

Suppose $f\in \mathcal{A}_{q,q'}(\Omega)$, $g\in \mathcal{A}_{\mathfrak{q}, \mathfrak{q}',r}(\Omega)$ and let ${\bf B}, {\bf C} : \Omega\to \mathbb H\cong \mathbb R^4$ be two conservative vector fields such that ${\bf B} f = D_{q,q'} f - {}^{\psi}\mathcal{D} [f ]$ and $g{\bf C} = D_{\mathfrak{q},\ma on $\Omega$.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 23 more