Dunkl shift operators and Bannai-Ito polynomials

TL;DR

This paper characterizes Dunkl shift operators that preserve polynomial spaces and reveals their eigenfunctions as Bannai-Ito polynomials, introducing new complementary polynomials and algebraic structures.

Contribution

It provides a comprehensive analysis of Dunkl shift operators with reflections, explicitly constructs Bannai-Ito polynomials, and introduces complementary BI polynomials as a new limit case.

Findings

Eigenfunctions of the operator are Bannai-Ito polynomials.

Explicit expression of BI polynomials in terms of Wilson polynomials.

Introduction of complementary BI polynomials as a new $q o -1$ limit.

Abstract

We consider the most general Dunkl shift operator $L$ with the following properties: (i) $L$ is of first order in the shift operator and involves reflections; (ii) $L$ preserves the space of polynomials of a given degree; (iii) $L$ is potentially self-adjoint. We show that under these conditions, the operator $L$ has eigenfunctions which coincide with the Bannai-Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal with respect to the action of the operator $L$. This allows to express the BI polynomials explicitly. We also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations of this algebra, we derive the structure and recurrence relations of the BI polynomials. We introduce new orthogonal polynomials - referred to as the complementary BI polynomials - as an alternative $q \to -1$ limit of the Askey-Wilson polynomials. These complementary BI polynomials lead to a new explicit expression for the BI polynomials in terms of the ordinary Wilson polynomials.