On Zeros of Functions of Mittag--Leffler Type

Abstract

As is well known, the asymptotics of zeros of functions of Mittag--Leffler type

$$E_\rho \left( {z;\mu } \right) = \sum\limits_{n = 0}^\infty {\frac{{z^n }}{{\Gamma \left( {\mu + {n \mathord{\left/ {\vphantom {n \rho }} \right. \kern-\nulldelimiterspace} \rho }} \right)}}} ,{\text{ }}\rho >0,{\text{ }}\mu \in \mathbb{C},$$

describes the behavior of zeros outside a disk of sufficiently large radius. In the paper we solve the problem of finding the number of zeros inside such a disk; this allows us to indicate the numeration of all zeros $E_\rho \left( {z;\mu } \right)$ that agrees with the asymptotics. We study the problem of the distribution of zeros of two functions that can be expressed in terms of $E_1 \left( {z;\mu } \right)$, namely of the incomplete gamma-function and of the error function.