Derivatives of entire functions and a question of Pólya

Abstract:

An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form \[ f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}\] where $a \geqslant 0,b$ and the ${z_n}$ are real, and ${\Sigma _n}z_n^{ - 2} < \infty$. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if $f,f’$, and $f''$ have only real zeros (f of finite order), then either f has the above form or f has one of the forms \[ f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})\] where a, b, c, and d are constants, b complex, c and d real.