Special monogenic polynomials - properties and applications

Abstract

In Clifford Analysis several different methods have been developed for constructing monogenic functions as series with respect to properly chosen homogeneous monogenic polynomials. Almost all these methods rely on sets of orthogonal polynomials with their origin in classical (real) Harmonic Analysis in order to obtain the desired basis of homogeneous polynomials. We use a direct and elementary approach to this problem and construct a set of homogeneous polynomials involving only products of a hypercomplex variable and its hypercomplex conjugate. The obtained set is an Appell set of monogenic polynomials with respect to the hypercomplex derivative. Its intrinsic properties and some applications are presented.