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

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