Consider the following expression: $$ f(s) = s^{n+1} \, \Phi(s^2,1,-1-\epsilon) - s^3 \Phi \left(s^2,1,-\frac{n}{2}-\epsilon \right) \, , $$ where $\Phi$ is the Lerch transcendent function (implemented in Maple as LerchPhi). Here $s \in [0,1]$. I was wondering whether there exists a technique that allows the analytical evaluation of the limit of $f(s)$ as $\epsilon \to 0$.
Actually, it can be shown that this emerges in the evaluation of the sum $$ \sum_{k=0}^{\frac{n}{2}-1} \frac{s^{2k+3}}{\frac{n}{2}-k} = s^{n+1} + \lim_{\epsilon\to 0} f(s) \, , $$ where $n$ is an even integer.
Any help or hints are most welcome!
We have $$f(s) = s^{n + 1} \sum_{k \geq n/2 - 1} \frac {s^{2 (k - n/2 + 1)}} {k - n/2 - \epsilon} - s^3 \sum_{k \geq 0} \frac {s^{2 k}} {k - n/2 - \epsilon} = \\ -\sum_{0 \leq k < n/2 - 1} \frac {s^{2 k + 3}} {k - n/2 - \epsilon} \xrightarrow {\epsilon \to 0} \sum_{0 \leq k < n/2 - 1} \frac {s^{2 k + 3}} {n/2 - k}.$$ A closed form can be obtained by using the identity $$\Phi(z, 1, a) = \frac 1 z \Phi \!\left( \frac 1 z, 1, 1 - a \right) + \pi (-z)^{-a} \csc \pi a, \\ |z| > 1, \;a \notin \mathbb Z$$ and then passing to the limit, which gives $$\lim_{\epsilon \to 0} f(s) = s^{n - 1} \Phi \!\left(\frac 1 {s^2}, 1, 2 \right) - s \Phi \!\left( \frac 1 {s^2}, 1, \frac n 2 + 1 \right).$$