An algebraic interpretation of the multivariate q -Krawtchouk polynomials

The multivariate quantum q -Krawtchouk polynomials are shown to arise as matrix elements of “ q -rotations” acting on the state vectors of many q -oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations, and recurrence relations. The extension to an arbitrary number of variables is presented.