I'm currently researching a general closedform formula, in terms of elementary functions, for functions that have the following type of partial fraction expansion: $$\frac{1}{x^{p}}+\sum_{n=1}^{+\infty}\left(\frac{1}{(xn)^{p}}+\frac{1}{(x+n)^{p}}\right). $$ I would appreciate any references to articles, books, or papers that investigate or provide general closedform formulas for such functions. Additionally, if you have insights or examples related to this type of expansion, please share them.
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These are Hurwitz zeta functions, $$\sum_{n=1}^{+\infty}\left(\frac{1}{(xn)^{p}}+\frac{1}{(x+n)^{p}}\right)=(1)^{p} \zeta (p,1x)+\zeta (p,1+x).$$ The Wikipedia entry for these special functions has quite an extensive list of formulas and pointers to the literature.
For integer $p$, as pointed out by Henri Cohen, this simplifies to $$\sum_{n=1}^{+\infty}\left(\frac{1}{(xn)^{p}}+\frac{1}{(x+n)^{p}}\right)=\frac{(1)^{p1} }{(p1)!}\frac{\partial ^{p1}}{\partial x^{p1}}\frac{\pi x \operatorname{cotan} (\pi x)1}{x}.$$

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$\begingroup$ I need references to articles that provide general closedform formula, in terms of elementary functions ($\cos,\ \sin$). $\endgroup$– L.LCommented Jun 20 at 13:18

1$\begingroup$ there is no representation in terms of elementary functions for general $p\in\mathbb{R}$, only for integer $p$. $\endgroup$ Commented Jun 20 at 13:45

1$\begingroup$ @L.L To expand higher derivatives of $\cot\pi x$ in terms of $\sin$ and $\cos$, see math.stackexchange.com/questions/5357/… . $\endgroup$ Commented Jun 21 at 7:24