Today I have encounter a series:
$$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$
where $u \not \in \Bbb{Z}$
. I have known a method to computer it (by Residue formula):
$$\int_{|z|=N+1/2}\frac{\pi \cot \pi z}{(u+z)^2}\text{dz}=-\frac{\pi^2}{\sin^2 \pi u}+\sum_{k=-N}^{N}\frac{1}{(u+k)^2}$$
where $-u$ is a pole of order 2. and $n \in \Bbb{Z}$ and $|n| \leq N$ are poles of order 1. because $$\left|\int \frac{\pi \cot \pi z}{(u+z)^2}\text{dz}\right|\leq \int_0^{2\pi}\pi\left|1+\frac{2}{e^{i2\pi z}-1}\right|\frac{1}{|((N+1/2)e^{i\varphi}+u)^2|}\text{d}\varphi \rightarrow 0$$ as $N \rightarrow +\infty$
Dose there exists some other way to computer it?
Best Answer
Consider the Weierstrass form of the Gamma function:
$$\frac{1}{\Gamma (x)}=xe^{\gamma x}\prod_{n\geq 1}\left(1+\frac{x}{n}\right)e^{-\frac{x}{n}}$$ Then:
$$\ln\Gamma (x)=-\ln x-\gamma x+\sum_{n\geq 1}\left[\frac{x}{n}-\ln \left(1+\frac{x}{n}\right)\right]$$
$$\frac{d}{dx}\ln\Gamma(x)=-\frac{1}{x}-\gamma+\sum_{n\geq 1}\left(\frac{1}{n}-\frac{1}{n+x}\right)$$
$$\begin{align*}\frac{d^2}{dx^2}\ln\Gamma (x)&=\frac{1}{x^2}+\sum_{n\geq 1}\frac{1}{(n+x)^2}\\[7pt]&=\sum_{n\geq 0}\frac{1}{(n+x)^2}\quad(1)\end{align*}$$
Note that by letting $x\to 1-x$ and $n\to-n-1$ we also have:
$$\frac{d^2}{dx^2}\ln\Gamma(1-x)=\sum_{n\leq -1}\frac{1}{(n+x)^2}\quad(2)$$
Combine $(1)$ and $(2):$
$$ \begin{align*}\sum_{\mathbb{Z}}\frac{1}{(n+x)^2}&=\frac{d^2}{dx^2}\ln\Gamma(1-x)+\frac{d^2}{dx^2}\ln\Gamma(x)\\&=\frac{d^2}{dx^2}\ln \Gamma(x)\Gamma(1-x)\\[7pt]&=\frac{d^2}{dx^2}\ln\pi\csc \pi x\quad(x\not\in\mathbb{Z})\\[7pt]&=\pi^2\csc^2\pi x\end{align*}$$