Bispectrality of the Complementary Bannai-Ito Polynomials

TL;DR

This paper introduces a family of operators with complementary Bannai-Ito polynomials as eigenfunctions, revealing their algebraic structure, spectral properties, and connections to other polynomial families in the Askey scheme.

Contribution

It constructs a new family of operators for CBI polynomials, explores their algebraic relations, and links them to dual -1 Hahn and para-Krawtchouk polynomials.

Findings

Operators with CBI eigenfunctions are characterized.

The algebra of CBI polynomials is a deformation of the Askey-Wilson algebra.

Connections between CBI, dual -1 Hahn, and para-Krawtchouk polynomials are established.

Abstract

A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a $q\rightarrow-1$ limit of the Askey-Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey-Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.