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 inR 4 ( orR 3 ) with values in H . This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ ofH 4 ( orH 3 ) 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 ofH 4 . Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.