# The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

## TL;DR

This paper explores the algebraic structure and multivariate extensions of Bannai-Ito and Racah polynomials, revealing their bispectrality, orthogonality, and connection to higher rank algebras through difference operators.

## Contribution

It introduces multivariate Bannai-Ito polynomials, extends the algebraic framework to higher ranks, and proves a conjecture relating these polynomials to the $q=1$ Bannai-Ito algebra.

## Key findings

- Realization of the higher rank $q$-Bannai-Ito algebra via difference operators.
- Extension of Bannai-Ito polynomials to multiple variables.
- Proof of a conjecture connecting multivariate Bannai-Ito polynomials to the algebra.

## Abstract

The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting the action of the algebra to the connection coefficients, we find a realization of $\mathcal{A}_n^q$ by means of difference operators. This provides an algebraic interpretation for the bispectrality of the multivariate $(-q)$-Racah polynomials, as was established in [Iliev, Trans. Amer. Math. Soc. 363 (3) (2011), 1577-1598]. Furthermore, we extend the Bannai-Ito orthogonal polynomials to multiple variables and use these to express the connection coefficients for the $q = 1$ higher rank Bannai-Ito algebra $\mathcal{A}_n$, thereby proving a conjecture from [De Bie et al., Adv. Math. 303 (2016), 390-414]. We derive the orthogonality relation of these multivariate Bannai-Ito polynomials and provide a discrete realization for $\mathcal{A}_n$.