The Lerch-transcendent as given in Mathworld is
$$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$
I'm fiddling with series of the form
$$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their Borel-transforms
$$ g_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n }{1 \over k!} $$
such that $g_n(z)$ is also the Borel-transform of $\Phi(z,n,1)$.
I want now work with $f_n(z)$ and $g_n(z)$ (and their derivatives) at integer values $n$ only and want improve some hypotheses (so far only gotten by numerical approximations) for $g_n(z)$ and its derivatives - of course there might exist (possibly simple) expressions for this already. So my question:
Q: Are there closed form expressions known for the Borel-transform for the Lerch-transcendent (or at least for the $g_n(z)$ in my definition at integer values $n$) ?
Additional background: I'm trying to understand the behaviour of $f_n(z)$ when it is divergent, for instance for negative $n$. The Borel-transform $g_n(z)$ is still convergent (entire in both parameters) and finding analytic expressions for them might help to describe $f_n(z)$ better for the divergent (or conditionally convergent) cases.
They are called polyexponentials, see http://arxiv.org/abs/0710.1332