Theq-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 rankq-Bannai-Ito algebraAnq. Lifting the action of the algebra to the connection coefficients, we find a realization ofAnqby 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 theq=1higher rank Bannai-Ito algebraAn, thereby proving a conjecture from De Bieet al. (Adv. Math. 303 (2016) 390-414). We derive the orthogonality relation of these multivariate Bannai-Ito polynomials and provide a discrete realization forAn.