Hermite polynomials on the plane

The space ${\cal P}_n$ of bivariate generalised Hermite polynomials of degree n is invariant under rotations. We exploit this symmetry to construct an orthonormal basis for ${\cal P}_n$ which consists of the rotations of a single polynomial through the angles ${\ell\pi\over n+1}$ , ℓ=0,...n. Thus we obtain an orthogonal expansion which retains as much of the symmetry of ${\cal P}_n$ as is possible. Indeed we show that a continuous version of this orthogonal expansion exists.