A quaternionic fractional Borel-Pompeiu type formula
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 $ψ-$hyperholomorphic functions i.e., null-solutions of the $ψ-$Fueter operator related to a so-called structural set $ψ$ 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 $ψ-$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 $ψ-$hyperholomorphic function theory and the related operator calculus.