This is probably not a research level question but I honestly don't know how/where to look for techniques to reconstruct a function from its asymptotic expansion.
The expansion I want to know about occurred in connection with another question I asked before (this one); it is $$ \sum_{n\geqslant1}\frac{B_{2n}}{2n!}(-z)^{1-n}=\frac1{12}+\frac{z^{-1}}{120}+\frac{z^{-2}}{504}+\frac{z^{-3}}{1440}+\frac{z^{-4}}{3168}+\frac{691z^{-5}}{3931200}+... $$
My question is whether this expansion is familiar to anybody (it looks so "classical"!) or if not, whether there are general methods to obtain some representations (series, integrals) of a function from its asymptotic expansion.
(NB) Everything would be clear to me with (2n)! instead of 2n!
A later update:
Inspired by the comment of @IgorKhavkine I applied a slight modification of the Borel summation trick to the original asymptotic expansion and obtained \begin{multline*} \sum_{n\geqslant1}\frac{B_{2n}}{2n!}(-z)^{1-n}=\sum_{n\geqslant1}\frac{B_{2n}}{2n!}(-z)^{1-n}\frac{\int\limits_0^\infty t^{n-\frac12}e^{-t}dt}{\Gamma\left(n+\frac12\right)}\\=\sum_{n\geqslant1}\frac{B_{2n}4^n}{2(2n)!\sqrt\pi}(-z)^{1-n}\int\limits_0^\infty t^{n-\frac12}e^{-t}dt\\=\frac{-z}{2\sqrt\pi}\int\limits_0^\infty\frac1{\sqrt t}\sum_{n\geqslant1}\frac{B_{2n}(-4t/z)^n}{(2n)!}e^{-t}dt\\=\frac{-z}{2\sqrt\pi}\int\limits_0^\infty\frac1{\sqrt t}\left(\frac{\sqrt{-4t/z}}{e^{\sqrt{-4t/z}}-1}-1+\frac12\sqrt{-4t/z}\right)e^{-t}dt\\=\sqrt{\frac{-z}\pi}\int\limits_0^\infty\frac{e^{-t}}{e^{2\sqrt{-t/z}}-1}dt+\frac z2+\frac12\sqrt{\frac{-z}\pi} \end{multline*} At the moment I do not care so much about legitimacy of the above transformations; what matters to me is whether the obtained function coincides with the one I mentioned in a comment below. Moreover the function of $z$ I obtained in the end does indeed have the desired asymptotic expansion, except that I failed to find any information about the integral I have arrived at. Thus my question now becomes
Can the integral $$\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$$ be evaluated in closed form using some known special functions?