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Mat. Zametki, 2000, Volume 68, Issue 5, Pages 710–724 (Mi mz992)  

This article is cited in 20 scientific papers (total in 20 papers)

On Zeros of Functions of Mittag-Leffler Type

A. M. Sedletskii

M. V. Lomonosov Moscow State University

Abstract: As is well known, the asymptotics of zeros of functions of Mittag-Leffler type
$$ E_\rho(z;\mu)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(\mu+n/\rho)},\qquad\rho>0,\quad\mu\in\mathbb C, $$
describes the behavior of zeros outside a disk of sufficiently large radius. In the paper we solve the problem of finding the number of zeros inside such a disk; this allows us to indicate the numeration of all zeros $E_\rho(z;\mu)$ that agrees with the asymptotics. We study the problem of the distribution of zeros of two functions that can be expressed in terms of $E_1(z;\mu)$, namely of the incomplete gamma-function and of the error function.

DOI: https://doi.org/10.4213/mzm992

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English version:
Mathematical Notes, 2000, 68:5, 602–613

Bibliographic databases:

UDC: 517.5
Received: 14.09.1999

Citation: A. M. Sedletskii, “On Zeros of Functions of Mittag-Leffler Type”, Mat. Zametki, 68:5 (2000), 710–724; Math. Notes, 68:5 (2000), 602–613

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. Popov, AY, “The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions”, Differential Equations, 38:5 (2002), 642  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Tikhonov, IV, “An inverse problem for a differential equation in a Banach space and the distribution of zeros of an entire Mittag-Leffler function”, Differential Equations, 38:5 (2002), 669  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Popov, AY, “Zeros distribution of Mittag-Leffler functions”, Doklady Mathematics, 67:3 (2003), 336  zmath  isi
    4. A. M. Sedletskii, “Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function”, Math. Notes, 75:3 (2004), 372–386  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. A. M. Sedletskii, “On the Zeros of Laplace Transforms”, Math. Notes, 76:6 (2004), 824–833  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Tikhonov, IV, “A generalized ward problem for abstract differential equations”, Differential Equations, 41:3 (2005), 340  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Hanneken J.W., Vaught D.M., Achar B.N.N., “The Number of Real Zeros of the Single Parameter Mittag-Leffler Function for Parameter Values Between 1 and 2”, Proceedings of the Asme International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol 6, Pts a-C, Amer Soc Mechanical Engineers, 2005, 1401–1405  isi
    8. Craven, T, “The Fox-Wright functions and Laguerre multiplier sequences”, Journal of Mathematical Analysis and Applications, 314:1 (2006), 109  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. A. M. Sedletskii, “Asymptotics of the Zeros of Degenerate Hypergeometric Functions”, Math. Notes, 82:2 (2007), 229–237  mathnet  crossref  crossref  mathscinet  isi  elib
    10. Hanneken J.W., Vaught D.M., Achar B.N.N., “Enumeration of the Real Zeros of the Mittag-Leffler Function E-Alpha(Z), 1 <Alpha < 2”, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, eds. Sabatier J., Agrawal O., Machado J., Springer, 2007, 15–26  crossref  mathscinet  zmath  isi  scopus  scopus
    11. A. V. Glushak, “Inverse problems for evolution equations with fractional integrals at boundary-value conditions”, Journal of Mathematical Sciences, 164:4 (2010), 518–530  mathnet  crossref  mathscinet  elib
    12. Chikrii, AA, “Optimization of game interaction of fractional-order controlled systems”, Optimization Methods & Software, 23:1 (2008), 39  crossref  mathscinet  zmath  isi  scopus  scopus
    13. A. V. Glushak, “On an Inverse Problem for an Abstract Differential Equation of Fractional Order”, Math. Notes, 87:5 (2010), 654–662  mathnet  crossref  crossref  mathscinet  isi
    14. 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
    15. Hanneken J.W., Achar B.N.N., Vaught D.M., Andrews J.M., Carr A.T., “Sums of the Zeros of the Single Parameter Mittag-Leffler Function with the Parameter Equal to One-Half”, Proceedings of Asme International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol 4, Pts a-C, Amer Soc Mechanical Engineers, 2010, 995–1001  isi
    16. 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
    17. Glushak A.V., Manaenkova T.A., “Pryamaya i obratnaya zadachi dlya abstraktnogo differentsialnogo uravneniya, soderzhaschego drobnye proizvodnye adamara”, Differentsialnye uravneniya, 47:9 (2011), 1294–1304  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    18. Jin B., Lazarov R., Lu X., Zhou Zh., “a Simple Finite Element Method For Boundary Value Problems With a Riemann-Liouville Derivative”, J. Comput. Appl. Math., 293 (2016), 94–111  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Aleroev T.S. Aleroeva H.T. Huang J. Tamm M.V. Tang Y. Zhao Yu., “Boundary value problems of fractional Fokker–Planck equations”, Comput. Math. Appl., 73:6 (2017), 959–969  crossref  mathscinet  isi  scopus
    20. Glushak A.V. Nekrasova I.V. Florinsky V.V. Yaduta A.Z., “Direct and Inverse Problem of Cauchy Type For the Fraction Equation With the Degeneration and Non-Loaded Operator Coefficient”, Dilemas Contemp.-Educ. Politica Valores, 6:SI (2019), 66  isi
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