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The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ isconsidered. The intermediate Casimir operators are shown to realize a$q$-deformation of the Bannai-Ito algebra. The Racah coefficients of$\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonalpolynomials that $q$-generalize the Bannai-Ito polynomials. The relationbetween these $q$-deformed Bannai-Ito polynomials and the$q$-Racah/Askey-Wilson polynomials is discussed.

The quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials