<s>[INST] Expressing $\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$ in closed form I want to express $$\sum_{n=-\infty}^\infty\dfrac{1}{z^3-n^3}$$ in closed form.
I know that $$\pi z\cot(\pi z)=1+2z^2\sum_{n=1}^\infty\dfrac{1}{z^2-n^2}$$ which looks close, but I don’t know how to use it.
 [/INST] Here is a closed form by Maple

$$ \frac{2}{3}\,{\frac {\pi \, \left( -\cos \left( 2\,\pi \,z \right) +\sqrt {3}
\sin \left( \pi \,z \right) \sinh \left( \pi \,z\sqrt {3} \right) +
\cos \left( \pi \,z \right) \cosh \left( \pi \,z\sqrt {3} \right) 
 \right) }{{z}^{2} \left( 2\,\sin \left( \pi \,z \right) \cosh \left( 
\pi \,z\sqrt {3} \right) -\sin \left( 2\,\pi \,z \right)  \right) }}.$$

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