On the Pólya-Wiman properties of Differential Operators

Abstract: Let $\phi(x)=\sum \alpha_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $\phi(D)^mf$ has only real zeros whenever $m\geq m_0$" if and only if "$\alpha_0=0$ or $2\alpha_0\alpha_2 - \alpha_1^2 <0$", and that if $\phi$ does not represent a Laguerre-P\'olya function, then there is a Laguerre-P\'olya function $f$ of genus $0$ such that for every positive integer $m$, $\phi(D)^mf$ represents a real entire function having infnitely many nonreal zeros.