  Lee, in a series of papers, described a unified formulation of the
statistical thermodynamics of ideal quantum gases in terms of the polylogarithm
functions, $\textup{Li}_{s} (z)$. It is aimed here to investigate the functions
$\textup{Li}_{s} (z),$ for $s = 0, -1, -2, ...,$ which are, following Lee,
referred to as the polypseudologarithms (or polypseudologs) of order $n$.
Various known results regarding polypseudologs, mainly obtained in widely
differing contexts and currently scattered throughout the literature, have been
brought together along with many new results and insights and they all have
been proved in a simple and unified manner. In addition, a new general explicit
closed-form formula for these functions involving the Carlitz--Scoville higher
tangent numbers has been established.
