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

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 ${\mathcal{A}}_n$.