I stumbled on the following suspected equality of infinite sums that I believe holds for $c \in \mathbb{R}$ and $x \in \mathbb{R} - \mathbb{Z}$:
$$\sum_{n=-\infty}^\infty \frac{\sin(2 \pi c (x - n))}{\pi (x - n)} = \sum_{n=-\infty}^\infty \operatorname{sign}(c - n) \cos(2 \pi n x)$$
I found this very interesting because for any fixed $c$, the summation on the right truncates to a finite sum of cosines. Furthermore, the result doesn't depend on the exact value of $c$ but only its floor $\lfloor c \rfloor$ and ceiling $\lceil c \rceil$, since these give us full information about $\operatorname{sign}(c - n)$ for all $n \in \mathbb{Z}$.
Wolfram Alpha didn't return anything useful, and I don't think I'm familiar with the tools that would be required to prove such an equality. This question is a generalization (by choosing $c = 1/2$) of a related question I found while searching for an answer: What is the sum over a shifted sinc function?
