Polypseudologarithms revisited

TL;DR

This paper provides a comprehensive and unified analysis of polypseudologarithms, including new formulas and insights, connecting scattered results and introducing a closed-form expression involving higher tangent numbers.

Contribution

It offers a unified treatment of polypseudologarithms for negative integer orders and introduces a new explicit formula involving Carlitz--Scoville higher tangent numbers.

Findings

Unified proofs of known polypseudolog results

New explicit closed-form formula involving higher tangent numbers

Enhanced understanding of polypseudologarithms in thermodynamics

Abstract

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.