On Zeros of Functions of Mittag-Leffler Type

A. M. Sedletskii

Matematičeskie zametki, Tome 68 (2000) no. 5, pp. 710-724

Résumé

As is well known, the asymptotics of zeros of functions of Mittag-Leffler type $$ E_\rho(z;\mu)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(\mu+n/\rho)},\qquad\rho>0,\quad\mu\in\mathbb C, $$ 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 $E_\rho(z;\mu)$ that agrees with the asymptotics. We study the problem of the distribution of zeros of two functions that can be expressed in terms of $E_1(z;\mu)$, namely of the incomplete gamma-function and of the error function.