Appell Bases for Monogenic Functions of Three Variables

Abstract

Monogenic (or hyperholomorphic) functions are well known in general Clifford algebras but have been little studied in the particular case \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}\). We describe for this case the collection of all Appell systems: bases for the finite-dimensional spaces of monogenic homogeneous polynomials which respect the operator \({D = \partial_{0} - \vec{\partial}}\). We prove that no purely algebraic recursive formula (in a specific sense) exists for these Appell systems, in contrast to the existence of known constructions for \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{4}}\) and \({\mathbb{R}^{4} \rightarrow \mathbb{R}^{4}}\). However, we give a simple recursive procedure for constructing Appell bases for \({\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}\) which uses the operation of integration of polynomials.