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

Abstract

At the negative integers, there is a simple relation between the Lerch $\Phi$ function and the polylogarithm. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Starting from that formula, we can deduce a simple closed formula for the Lerch $\Phi$ function at the negative integers, where the Stirling numbers are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $\Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we present a new formula for the Hurwitz zeta function at the positive integers using this novelty.