Geodesic
Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function
Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 405-420.

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We completely solve the problem of finding the number of positive and nonnegative roots of the Mittag-Leffler type function $$ E_\rho(z;\mu)=\sum_{n=0}^\infty \frac{z^n}{\Gamma(\mu+n/\rho)}, \qquad \rho>0, \qquad \mu\in\mathbb C, $$ for $\rho>1$ and $\mu\in\mathbb R$. We prove that there are no roots in the left angular sector $\pi/\rho\le|\arg z|\le\pi$ for $\rho>1$ and $1\le\mu1+1/\rho$. 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. 
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A. M. Sedletskii. Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function. Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 405-420. http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a7/

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