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On a sufficient condition for a Mittag-Leffler function to have real zeros only, and the P\'olya-Wiman properties of differential operators : 주어진 미탁-레플러 함수가 실근만을 갖기 위한 충분조건과 미분 연산자의 폴랴-위만 성질

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Authors

김민희

김영원
Major
자연과학대학 수리과학부
Issue Date
2016-02
Publisher
서울대학교 대학원
Keywords
Mittag-Leffler functionsP\'olya-Wiman Theoremzeros of polynomials and entire functionslinear differential operatorsLaguerre-P\'olya classHermite polynomialsDe Bruijn-Newman constant
Description
학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 김영원.
Abstract
In this dissertation, we study the distribution of zeros of entire functions.
First, we study the reality of zeros of Mittag-Leffler functions.
If $\alpha$ and $\beta$ are complex numbers with $\mathrm{Re}\ \alpha>0$, the Mittag-Leffler function $E_{\alpha,\beta}$ is defined by
\begin{equation*}
E_{\alpha,\beta} (z)=\sum_{k=0}^{\infty} \frac{z^k}{\Gamma{(\beta+\alpha k })}.
\end{equation*}
One of the most recent results on the zeros of the Mittag-Leffler functions is due to Popov and Sedletskii: if $\alpha > 2$ and $0 < \beta \leq 2\alpha - 1$ or if $\alpha > 4$ and $0 < \beta \leq 2\alpha$ then $E_{\alpha,\beta}(z)$ has only real zeros. We improve the result by showing that
if $\alpha\geq4.07$ and $0<\beta\leq 3\alpha$ then $E_{\alpha,\beta}(z)$ has only real zeros. %We also obtain some asymptotic results on the distribution of zeros of Mittag-Leffler functions in %the general case where is $\alpha >2$.

Second, we study the P\'olya-Wiman properties of differential operators.
Let $\phi(x)=\sum \alpha_n x^n$ 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 $m_0$ such that $\phi(D)^mf$ has only real zeros whenever
$m\geq m_0$'' if and only if $\alpha_0=0$ or $2\alpha_0\alpha_2 - \alpha_1^2 <0$'', and that if $\phi$ does not represent a
Laguerre-P\'olya function, then there is a Laguerre-P\'olya function
$f$ of genus $0$ such that for every positive integer $m$,
$\phi(D)^mf$ represents a real entire function having infinitely many
nonreal zeros.

Finally, we prove the identity
$$\sup\{\alpha\in\mathbb{R}:e^{\alpha D^2}\cos{ D}\ M^n \mbox{\ has real zeros only}\}=4{\lambda_n}^{-2},$$
where $M^n$ is the monic monomial of degree $n$, that is, $M^n (z)=z^n$, and $\lambda_n$ is the largest zero of the $2n$-th Hermite polynomial $H_{2n}$ given by
$$H_{2n}(z)=(2n)!\sum_{k=0}^{n}\frac{(-1)^k}{k!(2n-2k)!}(2z)^{2n-2k}.$$
Language
English
URI
https://hdl.handle.net/10371/121302
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)