A Borel-Pompeiu formula in a $(q,q^{\prime })$-model of quaternionic analysis

Abstract

The study of $\psi -$hyperholomorphic functions defined on domains in ${\mathbb{R}}^4$ with values in $\mathbb{H}$, namely null-solutions of the $\psi -$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 $(q,q^{\prime })-$calculus, also known as post quantum calculus, we introduce a deformation of the $\psi -$Fueter operator written in terms of suitable difference operators, which reduces to a deformed $q$ calculus when $q^{\prime }=1$. 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 $(q,q^{\prime })-$calculus theory.