Inspired by
I started to wonder about simpler series. I have seen similar questions to the following, but none had this special case explicitly. It is related to the q-digamma function. The answer is not the final two terms though as seen here. Source for partial sum formula. Integral representation of sech source. Second source for integral representation partial sum.
$$\mathrm{S_{sh}\mathop=^{def}\sum_{x\in\Bbb Z}sech(x)= \sum_{x\in\Bbb Z} \frac{2cosh(x)}{cosh(2x)+1}=1+2\sum_{x=1}^{\infty}\frac{2}{e^x+e^{-x}}=1+2i\lim_{n\to\infty} ψ_e\left(n-\frac{i\pi}2+1\right)-ψ_e\left(n-\frac{i\pi}2+1\right)-ψ_e\left(\frac{i\pi}2\right)+ ψ_e\left(-\frac{i\pi}2\right)}$$
Integral representation:
$$\mathrm{S_{sh}=\sum_{x\in\Bbb Z}\frac2\pi\int_0^\infty \frac{t^{\frac{2x i}{\pi}}}{t^2+1}dt=1+\frac1\pi\lim_{b\to\infty} \int_0^\infty\frac{1}{t^2+1}\sum_{x=0}^bt^\frac{2xi}\pi dt= 1+\frac1\pi\lim_{b\to\infty} \int_0^\infty\frac{t^\frac{2i(b+1)}{\pi}-1}{\left(t^\frac{2i}{\pi}-1\right)(t^2+1)}dt=3.14224265993564633914314598537…>\pi=3.1415..}$$
Jacobi theta function about page. Alternate forms. After a change of functions, @Gary found that the constant has the closed form of:
$$\mathrm{S_{sh}=\vartheta_3^2\left(\frac1e\right)}$$
Also from @Gary. As a bonus, this hyperbolic cosecant version has other forms, but this is the simplest closed form. The q-digamma term is about $\pi$:
$$\mathrm{\sum_{x\ge1}csch(x)=-ln\left(e^2-1\right)-ψ_{e^2}\left(\frac12\right)= 1.284423027303676524572857579841…}$$
Another interesting thing is the integral expression. How do I evaluate S in closed form as it most likely can be expressed in terms of the q-digamma function above? Also, how would one even derive the sum using the q digamma function? Please correct me and give me feedback!
Best Answer
If we expand $$ \operatorname{sech}n = e^{ - n} \frac{1}{{1 + e^{ - 2n} }} = e^{ - n} (1 - e^{ - 2n} + e^{ - 4n} - \cdots ) $$ and change the order of summation, we obtain $$ S=1 + 4\sum\limits_{m = 0}^\infty {\frac{{( - 1)^m }}{{e^{2m + 1} - 1}}} = \vartheta _3^2 (0,e^{ - 1} ), $$ where $$ \vartheta _3 (z,q) = \sum\limits_{n = - \infty }^{ \infty } {q^{n^2 } e^{2niz} } $$ is a Jacobi theta function. An alternative series is $$ S=\sum\limits_{n = 0}^\infty {r_2 (n)e^{ - n} } , $$ where $r_2(n)$ is the number of representations of $n$ by the sum of two squares, allowing zeros and distinguishing signs and order.