  Closed form, analytical results for the finite-temperature one-body density
matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of
particles obeying exclusion statistics are presented. As an application of our
general expressions, we consider the intermediate particle statistics arising
from the Gentile statistics, and compare its thermodynamic properties to the
Haldane fractional exclusion statistics. At low temperatures, the thermodynamic
quantities derived from both distributions are shown to be in excellent
agreement. As the temperature is increased, the Gentile distribution continues
to provide a good description of the system, with deviations only arising well
outside of the degenerate regime. Our results illustrate that the exceedingly
simple functional form of the Gentile distribution is an excellent alternative
to the generally only implicit form of the Haldane distribution at low
temperatures.
