Асимптотические фор мулы для нулей функци и типа Миттаг-Леффлера

Abstract

The paper is devoted to study the asymptotic behaviour of zerosz n of an entire function of Mittag-Leffler's type $$E_p (z;\mu ) = \sum\limits_{n = 0}^\infty {{{z^n } \mathord{\left/ {\vphantom {{z^n } {\Gamma (\mu + {n \mathord{\left/ {\vphantom {n \rho }} \right. \kern-\nulldelimiterspace} \rho }),\rho > 0,\mu \in C.}}} \right. \kern-\nulldelimiterspace} {\Gamma (\mu + {n \mathord{\left/ {\vphantom {n \rho }} \right. \kern-\nulldelimiterspace} \rho }),\rho > 0,\mu \in C.}}}$$ Asymptotic formulae are found which describe the values ofz . These formulae are accurate up to an infinitesimal quantity, whose order of magnitude is also estimated. The only exception is the caseρ=1/2, Reμ=3, where the value ofz 1/2 is described with an accuracy of O (1).