| Abstract: | We completely solve the problem of finding the number of positive and nonnegative roots of the Mittag-Leffler type function $E\rho (z;\mu ) = \sum\limits_{n = 0}^\infty {\frac{{z^n }}{{\Gamma (\mu + n/\rho )}}} , \rho >0, \mu \in \mathbb{C},$ for ρ > 1 and $\mu \in \mathbb{R}$ . We prove that there are no roots in the left angular sector $\pi /\rho \leqslant |\arg z| \leqslant \pi $ for ρ > 1 and 1≤µ<1 + 1/ρ. We consider the problem of multiple roots; in particular, we show that the classical Mittag-Leffler function E n(z;1) of integer order does not have multiple roots. |
