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A unified matrix approach to the representation of Appell polynomials

Lidia Aceto, Helmuth R. Malonek, Graça Tomaz

Abstract

In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. By this means we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations.

A unified matrix approach to the representation of Appell polynomials

Abstract

In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. By this means we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations.

Paper Structure

This paper contains 5 sections, 8 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Classical equivalent characterizations of Appell polynomials
  3. Appell polynomials: the matrix approach
  4. Some properties of Appell polynomials
  5. Conclusion

Key Result

Theorem 1

Let $H$ be the creation matrix defined by (H). If $G(x,t) \equiv f(t)e^{xt}$ is the generating function of an Appell sequence $\{p_n(x)\}_{n\geq 0}$, then the transfer matrix $M$ is a nonsingular matrix equal to $f(H).$

Theorems & Definitions (12)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • ...and 2 more