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I'm currently researching a general closed-form formula, in terms of elementary functions, for functions that have the following type of partial fraction expansion: $$\frac{1}{x^{p}}+\sum_{n=1}^{+\infty}\left(\frac{1}{(x-n)^{p}}+\frac{1}{(x+n)^{p}}\right). $$ I would appreciate any references to articles, books, or papers that investigate or provide general closed-form 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, $$\sum_{n=1}^{+\infty}\left(\frac{1}{(x-n)^{p}}+\frac{1}{(x+n)^{p}}\right)=(-1)^{-p} \zeta (p,1-x)+\zeta (p,1+x).$$ 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 $$\sum_{n=1}^{+\infty}\left(\frac{1}{(x-n)^{p}}+\frac{1}{(x+n)^{p}}\right)=\frac{(-1)^{p-1} }{(p-1)!}\frac{\partial ^{p-1}}{\partial x^{p-1}}\frac{\pi x \operatorname{cotan} (\pi x)-1}{x}.$$

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    $\begingroup$ $\cot(\pi x)$ and derivatives. $\endgroup$ Commented Jun 19 at 18:03
  • $\begingroup$ indeed, for integer $p$. $\endgroup$ Commented Jun 19 at 19:50
  • $\begingroup$ I need references to articles that provide general closed-form formula, in terms of elementary functions ($\cos,\ \sin$). $\endgroup$
    – L.L
    Commented Jun 20 at 13:18
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    $\begingroup$ 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
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    $\begingroup$ @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

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