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

Abstract

An integer-valued function is an entire function which maps the nonnegative integers 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 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.