With some Poisson summation manipulations (*credit: Michał Pacholski*) I have convinced myself of a closed form expression for this conditionally convergent series:


$$\sum\_{n=-\infty}^\infty \frac{e^{in\alpha}}{z+n}=\frac{2\pi i}{e^{i\alpha z}-e^{i(\alpha-2\pi) z}},\;\;\alpha\in(0,2 \pi),\;\;z\in\mathbb{C}\backslash\mathbb{Z}.$$


Mathematica returns a [Lerch zeta function](https://en.wikipedia.org/wiki/Lerch_zeta_function) (more precisely: a sum of two Hurwitz-Lerch transcendents), without further simplification. The above formula implies for the Lerch zeta function $L(a,b,s)=\sum\_{n=0}^\infty \frac{e^{2\pi ina}}{(n+b)^s}$ the reflection formula
$$L(a,b,1)-L(-a,-b,1)=\frac{1}{b}+\frac{2\pi i}{e^{2\pi iab}-e^{2\pi i(a-1) b}},\;\;a\in(0,1),\;\;b\in\mathbb{C}\backslash\mathbb{Z},$$
or more generally for $s\in\mathbb{N}$
$$L(a,b,s)+(-1)^{s}L(-a,-b,s)=\frac{1}{b^{s}}+\frac{(-1)^{s-1}}{(s-1)!}\frac{d^{s-1}}{db^{s-1}}\left(\frac{2\pi i}{e^{2\pi iab}-e^{2\pi i(a-1) b}}\right).$$


I would be pleased to learn if this is a known result for this special function, or a reference to a derivation in the literature.



Is this closed-form summation a special case of known Lerch zeta function formulas?
This is the Fourier series for the RHS, as a function of $\alpha\in (0,2\pi)$,
$$
f(\alpha)=\frac{2\pi i}{1-e^{-2\pi iz}}\, e^{-iz\alpha} .
$$
The series representation follows by computing the Fourier coefficients and noting that $f$ (as a function on the circle) is smooth away from $\alpha\equiv 0\bmod 2\pi$, so the Fourier series converges to the function.


