A quaternionic fractional Borel-Pompeiu type formula

TL;DR

This paper develops a fractional quaternionic Borel-Pompeiu formula, extending quaternionic analysis with fractional calculus to create a new framework for $ ext{ extpsi}$-hyperholomorphic functions and operator calculus.

Contribution

It introduces a fractional analogue of the Borel-Pompeiu formula for quaternionic functions, advancing the theory of fractional $ ext{ extpsi}$-hyperholomorphic functions.

Findings

Established a fractional Borel-Pompeiu formula in quaternionic analysis.

Extended the theory of $ ext{ extpsi}$-hyperholomorphic functions to fractional calculus.

Provided foundational tools for future fractional quaternionic operator theory.

Abstract

Quaternionic analysis relies heavily on results on functions defined on domains in $\mathbb R^4$ (or $\mathbb R^3$) with values in $\mathbb H$. This theory is centered around the concept of $\psi-$hyperholomorphic functions i.e., null-solutions of the $\psi-$Fueter operator related to a so-called structural set $\psi$ of $\mathbb H^4$. Fractional calculus, involving derivatives-integrals of arbitrary real or complex order, is the natural generalization of the classical calculus, which in the latter years became a well-suited tool by many researchers working in several branches of science and engineering. In theoretical setting, associated with a fractional $\psi-$Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analogue of Borel-Pompeiu formula as a first step to develop a fractional $\psi-$hyperholomorphic function theory and the related operator calculus.