On the Pólya-Wiman properties of Differential Operators

الأصل · EN

Let ϕ(x)=∑ αₙ xⁿ 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 m₀ such that ϕ(D)ᵐf has only real zeros whenever m≥ m₀" if and only if "α₀=0 or 2α₀α₂ - α₁² <0", and that if ϕ 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, ϕ(D)ᵐf represents a real entire function having infnitely many nonreal zeros.