On the Pólya-Wiman properties of Differential Operators

Min-Hee Kim; Young-One Kim

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arXiv · 1506.00350

On the Pólya-Wiman properties of Differential Operators

Min-Hee Kim·Young-One Kim

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Abstract

Let $ϕ(x)=\sum α_n x^n$ 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 $ϕ(D)^mf$ has only real zeros whenever $m\geq m_0$" if and only if "$α_0=0$ or $2α_0α_2 - α_1^2 <0$", 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.

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BibTeX RIS

Min-Hee Kim, Young-One Kim. 2015-06-01. On the Pólya-Wiman properties of Differential Operators. https://arxiv.org/abs/1506.00350

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