Representations of Some Classes of Quaternionic Hyperholomorphic Functions

In the algebra of complex quaternions \(\mathbb {H(C)}\) $\mathbb{H}(\mathbb{C})$ we consider the left– and right– \(\psi \) $\psi$ –hyperholomorphic functions, and left– \(\Lambda -\psi \) $\mathrm{\Lambda }-\psi$ –hyperholomorphic functions. We justify the transition in left– and right– \(\psi \) $\psi$ –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– \(\psi \) $\psi$ –hyperholomorphic functions, and representation of left– \(\Lambda -\psi \) $\mathrm{\Lambda }-\psi$ –hyperholomorphic functions.