A Bochner Theorem for Dunkl Polynomials

TL;DR

This paper extends Bochner's classical theorem to Dunkl operators, classifying all polynomial solutions and identifying limits of q-Jacobi polynomials as the only orthogonal families in this setting.

Contribution

It provides a classification of Dunkl-type operators with polynomial solutions, revealing their connection to q-Jacobi polynomial limits.

Findings

Classified Dunkl-type operators with polynomial solutions.

Identified limits of q-Jacobi polynomials as unique orthogonal families.

Extended Bochner's theorem to Dunkl operator framework.

Abstract

We establish an analogue of the Bochner theorem for first order operators of Dunkl type, that is we classify all such operators having polynomial solutions. Under natural conditions it is seen that the only families of orthogonal polynomials in this category are limits of little and big $q$-Jacobi polynomials as $q=-1$.