$q$-Rotations and Krawtchouk polynomials

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown to arise as matrix elements of unitary "$q$-rotation" operators expressed as $q$-exponentials in the $\mathcal{U}_{q}(sl_{2})$ generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine $q$-Krawtchouk polynomials, through a duality relation.