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.