The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states

An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the three-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions. Many of their properties are derived by exploiting the group-theoretic setting. The bivariate Tratnik polynomials of Krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters. It is explained how the approach generalizes naturally to ( d + 1) dimensions and allows us to interpret multivariate Krawtchouk polynomials as matrix elements of SO ( d + 1) unitary representations. Indications are given on the connection with other algebraic models for these polynomials.