Let $z \in \mathbb H$. Then can show by serie representation of $\pi \cot (\pi z)$ and differentiating those $k - 1$ times, that the relation $$\sum_{n \in \mathbb{Z}} \frac{1}{(z + n)^k} = \frac{(-i 2 \pi)^k}{(k - 1)!} \sum_{n = 1}^{\infty} n^{k - 1} e^{i 2 \pi n z}$$ holds for $k \in \mathbb N \setminus \{1\}$.
Can one generalize this to real $k > 1$ by substitution of $(k - 1)!$ by $\Gamma(k)$?
I have numerically validated this by high precision computing for a couple of values of $z$ and $k$.
If you know the relation between Polylogarthm function and Hurwitz zeta function the above equality can be approved easily.
LHS:
$ \frac{(-i 2 \pi)^k}{(k - 1)!} \sum_\limits{n = 1}^{\infty} n^{k - 1} e^{i 2 \pi n z}=\frac{(-i 2 \pi)^k}{\Gamma(k)} \sum_\limits{n = 1}^{\infty}\dfrac{(e^{i 2 \pi z})^n}{n^{ 1-k}}=\frac{(-i 2 \pi)^k}{\Gamma(k)} Li_{1-k}(e^{i 2 \pi z})$
The relation https://en.wikipedia.org/wiki/Polylogarithm :
$Li_s(w)=\dfrac{\Gamma(1-s)}{(2\pi)^{1-s}}\Big[i^{1-s}\zeta(1-s,\frac{1}{2}+\frac{\ln(-w)}{2\pi i})+i^{s-1}\zeta(1-s,\frac{1}{2}-\frac{\ln(-w)}{2\pi i})\Big]$
In our case:
$s=1-k$, $w=e^{i 2 \pi z}$ and using the definition of Hurwitz zeta function
$\zeta(p,q)=\sum\limits_{n=0}^\infty\dfrac{1}{(n+q)^p}$
we get the RHS of the equality.
(Sorry for the delayed answer)