I am looking for references that deal with the asymptotic expansions of sums of the form $$s(n)=\sum_{k=0}^n g(n,k)$$ using the (or similar to) following method.
We have the generating function $$f(z)=\sum_{n=0}^\infty s(n) z^n $$ so that by Cauchy's integral formula $$s(n)=\frac{1}{2\pi i}\oint_{(0+)}\frac{f(z)}{z^{n+1}}\ dz.$$ From here I will ideally take the radius of the contour to $+\infty$, and in the case the arc vanishes we can use a Hankel contour to integrate the asymptotic terms. Then, an application of Watson's lemma should yield the complete expansion for $s$.
For example see: $(1)$, $(2)$, $(3)$.
For some context, I worked on a problem last summer which dealt with the case $f(z)=\operatorname{Li}_{-1/2}^2 (z)$; this is $(1)$ above. I found this method quite powerful and was intrigued by the types of expansions yielded.
This summer, I am looking to expand my work and maybe make some generalizations for the case where $f$ is a polylogarithm or a $\chi$-function.
I think this article @Gary cited https://epubs.siam.org/doi/10.1137/0403019 in his post might be of use, but it costs $36 to view, which is slightly out of my budget range.
Any resource is welcome. Thank you.
