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 Anq. Lifting the action of the algebra to the connection coefficients, we find a realization of Anq 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 An, 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 An.