INFINITE SERIES INVOLVING \sech^2(n \pi s)

In this blog entry we will prove the following three results related to the Jacobi Theta function


\begin{align*} &\sum_{n=-\infty}^\infty n^2 e^{-\pi n^2 } =\frac{1}{4\pi^{3/4}\Gamma\left(\frac34 \right)}\\ & \sum_{n=-\infty}^\infty \frac{n \sinh(n \pi )}{\cosh^2(n \pi )} =\frac{1}{2\sqrt{\pi}\Gamma^2\left(\frac34 \right)}\\ & \frac{1}{2}\sum_{n=1}^\infty \frac{1-\tanh(n \pi )}{n} =\frac{\pi}{4}+\frac14\ln \pi-\frac54 \ln 2 - \ln \Gamma\left( \frac34\right) \end{align*}


Click here for the proof.


It relies on the Jacobi triple product (proved here)


\begin{align*} \sum_{n=-\infty}^{\infty} q^{n^{2}} z^{n}&=\prod_{n=1}^{\infty}\left(1+z q^{2 n-1}\right)\left(1+z^{-1} q^{2 n-1}\right)\left(1-q^{2 n}\right) \label{jacobi triple product} \end{align*}


And special value of Jacobi theta function (proved here)


\begin{align*} \sum_{n=-\infty}^\infty e^{-\pi n^2 } &=\frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34 \right)} \end{align*}

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