On a sufficient condition for a Mittag-Leffler function to have real zeros only, and the P\'olya-Wiman properties of differential operators

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 $$ H_{2n}(z)=(2n)!\sum_{k=0}^{n}\frac{(-1)^k}{k!(2n-2k)!}(2z)^{2n-2k}. $$