# Hermite polynomials on the plane

## Shayne Waldron

## Abstract:

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

**Keywords:**
bivariate Hermite polynomials,
Laguerre polynomials,
Zernike polynomials,
quadrature for trigonometric polynomials,
representation theory,
orthogonal expansions

**Math Review Classification:**
Primary 33C45, 33D50;
Secondary 06B15, 42C15

**Length:** 7 pages

**Last Updated:** 19 March 2007

## Availability: