  Many developments in Mathematics involve the computation of higher order
derivatives of Gaussian density functions. The analysis of univariate Gaussian
random variables is a well-established field whereas the analysis of their
multivariate counterparts consists of a body of results which are more
dispersed. These latter results generally fall into two main categories:
theoretical expressions which reveal the deep structure of the problem, or
computational algorithms which can mask the connections with closely related
problems. In this paper, we unify existing results and develop new results in a
framework which is both conceptually cogent and computationally efficient. We
focus on the underlying connections between higher order derivatives of
Gaussian density functions, the expected value of products of quadratic forms
in Gaussian random variables, and V-statistics of degree two based on Gaussian
density functions. These three sets of results are combined into an analysis of
non-parametric data smoothers.
