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Two characterizations of Sheffer-Dunkl sequences

Alejandro Gil Asensi, Judit Minguez Ceniceros

Abstract

Sheffer polynomials can be characterized using different Stieltjes integrals. These families of polynomials have been recently extended to the Dunkl context. In this way some classical operators as the derivative operator or the difference operator are replaced as analogous operators in the Dunkl universe. In this paper we establish two Stieltjes integrals that help us to characterize the Sheffer-Dunkl polynomials.

Two characterizations of Sheffer-Dunkl sequences

Abstract

Sheffer polynomials can be characterized using different Stieltjes integrals. These families of polynomials have been recently extended to the Dunkl context. In this way some classical operators as the derivative operator or the difference operator are replaced as analogous operators in the Dunkl universe. In this paper we establish two Stieltjes integrals that help us to characterize the Sheffer-Dunkl polynomials.
Paper Structure (11 sections, 4 theorems, 91 equations)

This paper contains 11 sections, 4 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Thorne Theorem for the Sheffer-Dunkl polynomials
  3. Sheffer Theorem for the Sheffer-Dunkl polynomials
  4. Examples
  5. Truncated polynomials
  6. Discrete truncated Appell-Dunkl polynomials
  7. Other Appell-Dunkl polynomials
  8. Bernoulli-Dunkl polynomials
  9. Bernoulli-Dunkl polynomials of the second kind
  10. Euler-Dunkl polynomials
  11. Boole-Dunkl polynomials

Key Result

Theorem 1

Let $g(t)$, $f(t)$ be two formal series as in eq:functions-D. Then a sequence $\{s_{n,\nu}(x)\}_{n=0}^{\infty}$ is the Sheffer-Dunkl sequence for the pair $(g(t),f(t))$ if and only if there exists a function $\alpha_{\nu}(x)$ of bounded variation on $(-\infty,\infty)$ with the following properties

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4