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

Abstract

We review the closed forms of the partial Fourier sums associated with $\HP_k(n)$ from a previous paper and create an asymptotic expression for $\HP(n)$ as a way to obtain formulae for the full Fourier series (if $|b|<1$, one obtains a surprising pattern, $\HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^k\zeta(k)b^{k-1}$). Finally, the derived Fourier series formulae are used to obtain a formula for the Lerch transcendent function, $\Phi(e^z,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, at the positive integers $k$.