  We show that the recently discovered thermodynamic equivalence between
noninteracting Bose and Fermi gases in two dimensions, and between
one-dimensional Bose and Fermi systems with linear dispersion, both in the
grand-canonical ensemble, are special cases of a larger class of equivalences
of noninteracting systems having an energy-independent single-particle density
of states. We also conjecture that the same equivalence will hold in the
grand-canonical ensemble for any noninteracting quantum gas with a discrete
ladder-type spectrum whenever $\sigma \Delta / N k_{\rm B} T$ is small, where
$N$ is the average particle number and $\sigma$ its standard deviation,
$\Delta$ is the level spacing, $k_{\rm B}$ is Boltzmann's constant, and $T$ is
the temperature.
