A recently asked question (linked here) deals with the remarkable identity $$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$ where $\mathrm{sinc}(x)=\sin(x)/x$.

It is easy to construct functions $f$ other than $\mathrm{sinc}(x)$ such that $\sum_{n\in\mathbb Z} f(n+x)$ is constant for all real $x$: define $f$ outside of $[0,1)$ to ensure convergence and then let $f(x)=C-\sum_{n\in\mathbb Z\setminus\{0\}}f(n+x)$ for $x\in[0,1)$. I wonder, however, whether there are *analytic* functions other than $\mathrm{sinc}(x)$ with this property? The set of such functions is a vector space over the complex numbers; is it finite-dimensional? If so, what is its dimension?