On the Pólya-Wiman properties of differential operators

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• 作者：Min-Hee Kim^a ; ^{alsgml01@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Young-One Kim^b ; ^{kimyone@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Pó ; lya&ndash ; Wiman theorem ; Zeros of polynomials and entire functions ; Linear differential operators ; Laguerre&ndash ; Pó ; lya class ; Hermite polynomials ; Mittag&ndash ; Leffler functions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1091-1105
• 全文大小：391 K

文摘

Let ϕ(x)=∑α_nxⁿ$\varphi (x)=\sum {\alpha }_n{x}^n$ 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 m₀$m_0$ such that ebbaf84515e315cc" title="Click to view the MathML source">ϕ(D)^mf$\varphi {(D)}^{m}f$ has only real zeros whenever m≥m₀$m\ge m_0$” if and only if “α₀=0${\alpha }_0=0$ or $2{\alpha }_0{\alpha }_2-{\alpha }_1^2<0$”, 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  , ebbaf84515e315cc" title="Click to view the MathML source">ϕ(D)^mf$\varphi {(D)}^{m}f$ represents a real entire function having infinitely many nonreal zeros.