Lerch's $\Phi$ and the Polylogarithm at the Positive Integers

TL;DR

This paper explores the Fourier series representations related to the Lerch transcendent and polylogarithm functions at positive integers, providing new formulas and asymptotic expressions.

Contribution

It introduces novel Fourier series formulas for the Lerch transcendent and polylogarithm at positive integers, along with asymptotic expressions for harmonic sums.

Findings

Derived Fourier series formulas for $ ext{Li}_k(e^m)$ and $ ext{Φ}(e^m,k,b)$

Established asymptotic expressions for harmonic sums $HP(n)$

Identified patterns in Fourier series when $|b|<1$

Abstract

We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^k\zeta(k)b^{k-1}$). Finally, we use the found Fourier series formulae to obtain the values of the Lerch transcendent function, $\Phi(e^m,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, at the positive integers $k$.