Integer-valued analytic functions in a half-plane
In this paper the following result is proved: Theorem: Let \(A\in\mathbb{R}\) and \(M> 0\) and let \(f\) be analytic in the half-plane \(H= \{z: \text{Re\,}z\geq A\}\), such that \[ f(n)\in\mathbb{Z}\text{ for all }n\in\mathbb{Z}\text{ with }n\geq A. \] Assume that \[ |f(z)|= O(|z|^M 2^{|z|})\quad\text{as }|z|\to\infty,\;z\in H. \] Then there exist polynomials \(P_1\), \(P_2\) with \[ f(z)= P_1(z)+ P_2(z) 2^z. \] The same result for entire functions was proved by \textit{G. Pólya} [Über ganze ganzwertige Functionen, Gött. Nachr. 1920, 1--10 (1920; JFM 47.0299.02)].
scientific article; zbMATH DE number 5281680