  Using a different approach, we derive integral representations for the
Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$,
the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,
$\Phi(e^m,-k,b)$), that coincide with their Abel-Plana expressions. A slight
variation of the approach leads to different formulae. We also present the
relations between each of these functions and their partial sums. It allows one
to figure, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$
(when $k$ is a positive integer, we obtain a finite Taylor series, which is
nothing but the Faulhaber formula). The method used requires evaluating the
limit of $\Phi\left(e^{2\pi i\,x},-2k+1,n+1\right)+\pi i\,x\,\Phi\left(e^{2\pi
i\,x},-2k,n+1\right)/k$ when $x$ goes to $0$, which in itself already makes for
an interesting problem.
