I've been messing around with some things after looking at Jacobi Theta Functions and I happened to stumble upon these unexpectedly (at least to me) nice Laurent expansion and integral representation of a curious sum:
$$ \begin{split} \sum_{n=0}^\infty e^{-\sqrt{n}t} &= \frac12+\frac1{\sqrt{\pi}}\int_0^\infty\coth\left(\frac{t^2}{8x^2}\right)e^{-x^2}dx \\ &= \frac2{t^2}+\frac12-\frac1{\sqrt{\pi}}\sum_{n=1}^\infty \sin\left(\frac{\pi k}4\right) \frac{(-t)^k\zeta(1+\frac k2)}{(8\pi)^{k/2}\Gamma(\frac{1+k}{2})} \end{split} $$
Has this series been studied somewhere? Is there literature that I could possibly read on it? I don't even know what to call the thing, much less how to search for resources on it online.
