  At the negative integers, there is a simple relation between the Lerch $\Phi$
function and the polylogarithm. Starting from that relation and a formula for
the polylogarithm at the negative integers known from the literature, we can
deduce a simple closed formula for the Lerch $\Phi$ function at the negative
integers, where the Stirling numbers of the second kind 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 extend formulae for the Hurwitz zeta function only
valid at the positive integers to the complex half-plane using this novelty.
