On the Pólya-Wiman properties of Differential Operators

2015/06/01 by Min-Hee Kim, Kim, Min-Hee, Young-One Kim +1
Mathematics · #30C15 #30D15 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Functional Equations Stability Results #Iterative Methods for Nonlinear Equations #math.CV #msc:30C15 #msc:30D15

paper · pdf · doi:10.48550/arxiv.1506.00350

16 pages

arxiv created 2015/06/01 · openalex publication_date 2015/06/01 · arxiv updated 2015/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ϕ(x)=∑ α_n 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.

Citations

• The Zeros of Derivatives of Entire Functions and the Pólya-Wiman Conjecture

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