On the Polya-Wiman properties of differential operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:434

Article

Kim, Young-One^1,2 

[1] Seoul Natl Univ, Dept Math Sci, Seoul 151747, South Korea

[2] Seoul Natl Univ, Res Inst Math, Seoul 151747, South Korea

关键词: Polya-Wiman theorem;    Zeros of polynomials and entire functions;    Linear differential operators;    Laguerre-Polya class;    Hermite polynomials;    Mittag-Leffler functions;   

DOI  :  10.1016/j.jmaa.2015.09.056

来源: Elsevier

【 摘 要 】

Let phi(x) = Sigma 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 phi(D)(m) f has only real zeros whenever m >= m(0) if and only if alpha(0) = 0 or 2 alpha(0)alpha(2) - alpha(2)(1) < 0, and that if phi does not represent a Laguerre-Polya function, then there is a Laguerre-Polya function f of genus 0 such that for every positive integer m, phi(D)(m) f represents a real entire function having infinitely many nonreal zeros. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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