  We investigate the particle and kinetic energy densities of harmonically
trapped fermion gases at zero temperature in arbitrary dimensions. We derive
analytically a differential equation connecting these densities, which so far
have been proven only in one or two dimensions, and give other interesting
relations involving several densities or the particle density alone. We show
that in the asymptotic limit of large particle numbers, the densities go over
into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to
be used in the TF densities is identified uniquely. We derive an analytical
expansion for the remaining oscillating parts and obtain very simple closed
forms for the leading-order oscillating densities. Finally, we show that the
simple TF functional relation $\tau_{TF}[\rho]$ between kinetic and particle
density is fulfilled also for the asymptotic quantum densities $\tau(r)$ and
$\rho(r)$ including their leading-order oscillating terms.
