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Matrix approach to hypercomplex Appell polynomials

Lidia Aceto, Helmuth Robert Malonek, Graça Tomaz

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.

Matrix approach to hypercomplex Appell polynomials

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.

Paper Structure

This paper contains 4 sections, 1 theorem, 40 equations.

Table of Contents

  1. Introduction
  2. Basic concepts and notation
  3. Sequences of monogenic hypercomplex Appell polynomials
  4. Special families of monogenic Appell polynomials

Key Result

Theorem 3.1

Suppose $\tilde{c}_0 \neq 0$ is a given real number. The polynomials in (vetfi) are monogenic if their remaining coefficients satisfy the conditions

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.1
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 4 more