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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    A  n  q   . Lifting the action of the algebra to the connection coefficients, we find a realization of    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    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    A  n   .

The q ‐Bannai–Ito algebra and multivariate ( − q ) ‐Racah and Bannai–Ito polynomials