Matrix approach to hypercomplex Appell polynomials

TL;DR

This paper extends a matrix representation approach for real Appell polynomials to hypercomplex and Clifford algebra contexts, enabling new solutions to generalized Cauchy-Riemann systems in higher dimensions.

Contribution

It generalizes the matrix approach to homogeneous Appell polynomials in hypercomplex spaces, linking them to solutions of generalized Cauchy-Riemann systems.

Findings

Matrix approach applies to hypercomplex Appell polynomials.

Constructs solutions to generalized Cauchy-Riemann systems.

Enhances polynomial approximation techniques in hypercomplex analysis.

Abstract

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy-Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras.