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College of Natural Sciences (자연과학대학)
Dept. of Mathematical Sciences (수리과학부)
Theses (Ph.D. / Sc.D._수리과학부)

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

- Authors

- Advisor
- 김영원

- Major
- 자연과학대학 수리과학부

- Issue Date
- 2016-02

- Publisher
- 서울대학교 대학원

- Keywords
- Mittag-Leffler functions ; P\'olya-Wiman Theorem ; zeros of polynomials and entire functions ; linear differential operators ; Laguerre-P\'olya class ; Hermite polynomials ; De 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

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