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.
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Sedletskii, A.M. Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function. Mathematical Notes 75, 372–386 (2004). https://doi.org/10.1023/B:MATN.0000023316.90489.fe
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DOI: https://doi.org/10.1023/B:MATN.0000023316.90489.fe