Representations of Some Classes of Quaternionic Hyperholomorphic Functions

In the algebra of complex quaternions \(\mathbb {H(C)}\) we consider the left– and right–\(\psi \)–hyperholomorphic functions, and left–\(\Lambda -\psi \)–hyperholomorphic functions. We justify the transition in left– and right–\(\psi \)–hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan’s basis we find the solution of Cauchy–Fueter equation. By the same method we find representations of left– and right–\(\psi \)–hyperholomorphic functions, and representation of left–\(\Lambda -\psi \)–hyperholomorphic functions.