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A "missing" family of classical orthogonal polynomials

Luc Vinet, Alexei Zhedanov

TL;DR

The paper identifies a missing family of classical orthogonal polynomials as the q=-1 limit of little q-Jacobi polynomials, showing they are governed by a Dunkl-type operator that preserves polynomial spaces and yields explicit polynomial eigenfunctions. It establishes a full Dunkl-classical structure with μ=α/2, connects these polynomials to symmetric Jacobi via Christoffel/Geronimus transforms, and realizes the Askey-Wilson algebra in the q=-1 regime. In the α=0 case, it reveals a surprising square-root relation between the Dunkl operator and a classical Sturm-Liouville operator, with a Schrödinger interpretation and potential generalizations through Darboux factorizations. Together, these results broaden the landscape of classical orthogonal polynomials and highlight deep connections to algebraic and quantum-mechanical structures.

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 $q=-1$.

A "missing" family of classical orthogonal polynomials

TL;DR

The paper identifies a missing family of classical orthogonal polynomials as the q=-1 limit of little q-Jacobi polynomials, showing they are governed by a Dunkl-type operator that preserves polynomial spaces and yields explicit polynomial eigenfunctions. It establishes a full Dunkl-classical structure with μ=α/2, connects these polynomials to symmetric Jacobi via Christoffel/Geronimus transforms, and realizes the Askey-Wilson algebra in the q=-1 regime. In the α=0 case, it reveals a surprising square-root relation between the Dunkl operator and a classical Sturm-Liouville operator, with a Schrödinger interpretation and potential generalizations through Darboux factorizations. Together, these results broaden the landscape of classical orthogonal polynomials and highlight deep connections to algebraic and quantum-mechanical structures.

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 -Jacobi polynomials in the limit . We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for .

Paper Structure

This paper contains 7 sections, 2 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Limit of the little $q$-Jacobi polynomials as $q \to -1$
  3. Different forms of $R$-differential equation
  4. Relation with the symmetric Jacobi polynomials
  5. The Dunkl-classical property
  6. Askey-Wilson algebra relations for the little -1 Jacobi polynomials
  7. The special case $\alpha=0$ and a "square root" of the Schrödinger operator

Key Result

Proposition 1

The little -1 Jacobi polynomials $P_n^{(-1)}(x)$ satisfy the Dunkl-classical property (TPP) with $\mu=\alpha/2$, where the polynomials $\tilde{P}_n(x)$ are again little -1 Jacobi polynomials with parameters $(\alpha, \beta+2)$.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2