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Mat. Zametki, 2004, Volume 75, Issue 3, Pages 405–420 (Mi mz44)  

This article is cited in 6 scientific papers (total in 6 papers)

Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function

A. M. Sedletskii

M. V. Lomonosov Moscow State University

Abstract: We completely solve the problem of finding the number of positive and nonnegative roots of the Mittag-Leffler type function
$$ E_\rho(z;\mu)=\sum_{n=0}^\infty \frac{z^n}{\Gamma(\mu+n/\rho)}, \qquad \rho>0, \qquad \mu\in\mathbb C, $$
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 $1\le\mu<1+1/\rho$. 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.

DOI: https://doi.org/10.4213/mzm44

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English version:
Mathematical Notes, 2004, 75:3, 372–386

Bibliographic databases:

UDC: 517.5
Received: 24.10.2002

Citation: A. M. Sedletskii, “Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function”, Mat. Zametki, 75:3 (2004), 405–420; Math. Notes, 75:3 (2004), 372–386

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Pskhu, “On the real zeros of functions of Mittag-Leffler type”, Math. Notes, 77:4 (2005), 546–552  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. A. Yu. Popov, “On the number of real eigenvalues of a certain boundary-value problem for a second-order equation with fractional derivative”, J. Math. Sci., 151:1 (2008), 2726–2740  mathnet  crossref  mathscinet  zmath
    3. Rogosin S., Koroleva A., “INTEGRAL REPRESENTATION OF THE FOUR-PARAMETRIC GENERALIZED MITTAG-LEFFLER FUNCTION”, Lith Math J, 50:3 (2010), 337–343  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. A. Yu. Popov, A. M. Sedletskii, “Distribution of roots of Mittag-Leffler functions”, Journal of Mathematical Sciences, 190:2 (2013), 209–409  mathnet  crossref  mathscinet  zmath
    5. Hanneken J.W., Achar B.N.N., Vaught D.M., “An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function”, Adv. Math. Phys., 2013, 421685  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Bhalekar S., Patil M., “Singular Points in the Solution Trajectories of Fractional Order Dynamical Systems”, Chaos, 28:11 (2018), 113123  crossref  mathscinet  zmath  isi  scopus
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