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Shall we get the closed-form of $$ S(\gamma) = \sum_{n=0}^{\infty}\frac{1}{\cosh[(2n+1)\pi^2/\gamma]} $$ Any hint?

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    Have you any reason to think that a closed form could exist ? – 2017-02-19
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    If the summation were from $-\infty$ to $\infty$, one could express the answer in terms of elliptic functions. For $k=0,\ldots,\infty$, I doubt one can get something reasonable. – 2017-02-19
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    (unless you accept the answer written in terms of $q$-hypergeometric series) – 2017-02-19
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    @Startwearingpurple: So, what's the express when the summation is from -$\infty$ to $\infty$? Thanks. – 2018-08-09

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Contradicting my 1.5-year old comment: for $\Re a>0$, we have $$\sum_{n=0}^{\infty}\frac{1}{\cosh(2n+1)a}=\frac12\left[\vartheta_2\left(0|e^{-2a}\right)\right]^2,$$ where $\vartheta_2(z|q)$ denotes the Jacobi theta function. For the proof, one may use this formula and few other identities for elliptic functions.