The quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials

TL;DR

This paper explores the Racah problem for the quantum superalgebra (1|2), revealing a q-deformation of the Bannai-Ito algebra and introducing a new family of q-orthogonal polynomials that generalize Bannai-Ito polynomials.

Contribution

It introduces a q-analogue of the Bannai-Ito algebra and explicitly computes Racah coefficients using new q-orthogonal polynomials that extend Bannai-Ito polynomials.

Findings

The intermediate Casimir operators realize a q-deformation of the Bannai-Ito algebra.

Explicit Racah coefficients are expressed in terms of q-orthogonal polynomials.

The new q-Bannai-Ito polynomials relate to q-Racah and Askey-Wilson polynomials.

Abstract

The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal polynomials that $q$-generalize the Bannai-Ito polynomials. The relation between these $q$-deformed Bannai-Ito polynomials and the $q$-Racah/Askey-Wilson polynomials is discussed.