Generating functions for the Bannai-Ito polynomials

TL;DR

This paper derives the generating function for Bannai-Ito polynomials by connecting them to Racah coefficients of the rak{osp}(1|2) Lie superalgebra, using Dunkl oscillator realizations.

Contribution

It provides a new derivation of the Bannai-Ito polynomial generating function via Lie superalgebra representation theory and Dunkl oscillators.

Findings

Explicit generating function for Bannai-Ito polynomials derived.

Connection established between Bannai-Ito polynomials and rak{osp}(1|2) Racah coefficients.

Method offers a new algebraic approach to these polynomials.

Abstract

The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $\mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a realization of the recoupling problem in terms of three Dunkl oscillators.