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The symmetric Dunkl-classical orthogonal polynomials revisited

Khalfa Douak

Abstract

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach applied in connection with the Dunkl operator. The main aim of this technique is to determine the recurrence coefficients first and foremost. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and also confirm that only two families of orthogonal polynomials, that is, the generalized Hermite polynomials and the generalized Gegenbauer polynomials, belong to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied.

The symmetric Dunkl-classical orthogonal polynomials revisited

Abstract

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach applied in connection with the Dunkl operator. The main aim of this technique is to determine the recurrence coefficients first and foremost. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and also confirm that only two families of orthogonal polynomials, that is, the generalized Hermite polynomials and the generalized Gegenbauer polynomials, belong to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied.
Paper Structure (3 sections, 4 theorems, 96 equations)

This paper contains 3 sections, 4 theorems, 96 equations.

Table of Contents

  1. Introduction and motivation
  2. Construction of the symmetric $T_\mu$-classical orthogonal polynomials
  3. Applications

Key Result

Theorem 1.2

Chih2 A necessary and sufficient condition for the existence of an orthogonal polynomials sequence (OPS) $\{P_n\}_{n\geqslant0}$ with respect to $u_0$ is $\Delta_n=\det((u_0)_{i+j})_{i,j=0}^n\ne0$, for $n=0, 1, 2, \ldots$. The functional $u_0$ is then said to be regular.

Theorems & Definitions (6)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2