Closed formula for variation of Fourier series of Bernoulli polynomials

Question

The Fourier series for the periodic Bernoulli polynomials $$ \sum_{k \in \mathbb{Z}-\{0\}} \frac{e^{2\pi ikx}}{k^n} = - \frac{(2\pi i)^n}{n!} P_n(x), \hspace{0.5cm} n \geq 1 $$ is well known. I am wondering if there exists an explicit (non-recursive) evaluation of variant $$ \sum_{k \in \mathbb{Z}} \frac{e^{2\pi ikx}}{(k+a)^n} $$ where $a$ is not an integer. I have tried using the formula $$ \sum_{k \in \mathbb{Z}} \frac{e^{2\pi ikx}}{k+a} = \frac{\pi e^{\pi i(1-2x)a}}{\sin(\pi a)} = \frac{2\pi i e^{-2\pi iax}}{1-e^{-2\pi ia}}, \hspace{0.5cm} 0<x<1, \; a \notin \mathbb{Z} $$ and differentiating $n-1$ times, but I cannot seem to find a nice closed expression in the end. Any help is appreciated.