On some structural sets and a quaternionic (φ, ψ)-hyperholomorphic function theory

Abstract

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field H. It relies heavily on results on functions defined on domains in R4 or R3 with values in H. This theory is centred around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of H4 or H3 respectively. The main goal of this paper is to develop the nucleus of the (φ,ψ)-hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy-Riemann operators associated to a pair φ,ψ of structural sets of H4. Following a matrix approach, a generalized Borel-Pompeiu formula and the corresponding Plemelj-Sokhotzki formulae are established.