Integer--valued functions, Hurwitz functions and related topics: a survey

Abstract

An integer--valued function is an entire function which maps the nonnegative integers $\mathbb N$ to the integers. An example is $2^z$. A Hurwitz function is an entire function having all derivatives taking integer values at $0$. An example is ${\mathrm e}^z$. Lower bound for the growth order of such functions have a rich history. Many variants have been considered: for instance, assuming that the first $k$ derivatives at the integers are integers, or assuming that the derivatives at $k$ points are integers. These as well as and many other variants have been considered. We survey some of them.