On the Pólya-Wiman properties of Differential Operators

Abstract

Let $\varphi (x)=\sum {\alpha }_n{x}^n$ 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 $\varphi (D{)}^{m}f$ has only real zeros whenever $m\ge m_0$" if and only if "${\alpha }_0=0$ or $2{\alpha }_0{\alpha }_2-{\alpha }_1^2<0$", and that if $\varphi$ does not represent a Laguerre-Pólya function, then there is a Laguerre-Pólya function $f$ of genus $0$ such that for every positive integer $m$, $\varphi (D{)}^{m}f$ represents a real entire function having infnitely many nonreal zeros.

16 pages