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

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.