A "missing" family of classical orthogonal polynomials

TL;DR

This paper introduces a new family of classical orthogonal polynomials linked to Dunkl-type operators, derived from little $q$-Jacobi polynomials at $q=-1$, and explores their algebraic properties.

Contribution

It identifies and analyzes a novel family of orthogonal polynomials satisfying differential eigenvalue problems and connects them to the Askey-Wilson algebra at $q=-1$.

Findings

Polynomials satisfy a Dunkl-type eigenvalue problem.

They can be obtained as a limit of little $q$-Jacobi polynomials at $q=-1$.

They realize the Askey-Wilson algebra for $q=-1$.

Abstract

We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.