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Volume 64, Issue 1


Convergence of series in three parametric Mittag-Leffler functions

Jordanka Paneva-Konovska
  • Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 8, bul. Kliment Ohridski, BG-1000, Sofia, Bulgaria
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Published Online: 2014-03-06 | DOI: https://doi.org/10.2478/s12175-013-0188-0


In this paper we consider a family of 3-index generalizations of the classical Mittag-Leffler functions. We study the convergence of series in such functions in the complex plane. First we find the domains of convergence of such series and then study their behaviour on the boundaries of these domains. More precisely, Cauchy-Hadamard, Abel, Tauber and Littlewood type theorems are proved as analogues of the classical theorems for the power series.

MSC: Primary 33E12, 40E05, 40A30; Secondary 30B30, 40G10

Keywords: Mittag-Leffler function and its generalizations; series in special functions in complex domain; Cauchy-Hadamard; Abel; Tauber and Littlewood type theorems; entire functions; summation of divergent series

  • [1] GORENFLO, R.— MAINARDI, F.: On Mittag-Leffler function in fractional evolution processes, J. Comput. Appl. Math. 118 (2000), 283–299. http://dx.doi.org/10.1016/S0377-0427(00)00294-6CrossrefGoogle Scholar

  • [2] DZRBASHJAN, M. M.: Integral Transforms and Representations in the Complex Domain, Nauka, Moskow, 1966 (Russian). Google Scholar

  • [3] DZRBASHJAN, M. M.: Interpolation and Spectrum Expansions Associated with Fractional Differential Operators, IM, AAS & Univ. of Yerevan, Yerevan, 1983 (Russian). Google Scholar

  • [4] Higher Transcendental Functions 1–3 (A. Erdélyi et al, eds), McGraw-Hill, New York-Toronto-London, 1953–1955. Google Scholar

  • [5] HARDY, G.: Divergent Series, Oxford University Press, Oxford, 1949. Google Scholar

  • [6] KILBAS, A. A.— KOROLEVA, A. A.— ROGOSIN, S. V.: Multi-parametric Mittag-Leffler functions and their extension, Fract. Calc. Appl. Anal. 16 (2013), 378–404. Web of ScienceGoogle Scholar

  • [7] KILBAS, A. A.— SAIGO, M.— SAXENA, R. K.: Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct. 15 (2004), 31–49. http://dx.doi.org/10.1080/10652460310001600717CrossrefGoogle Scholar

  • [8] KILBAS, A. A.— SRIVASTAVA, H. M.— TRUJILLO, J. J.: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar

  • [9] KIRYAKOVA, V.— AL-SAQABI, B.: Transmutation method for solving Erdélyi-Kober fractional differintegral equations, J. Math. Anal. Appl. 211 (1997), 347–364. http://dx.doi.org/10.1006/jmaa.1997.5469CrossrefGoogle Scholar

  • [10] KIRYAKOVA, V.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000), 241–259. http://dx.doi.org/10.1016/S0377-0427(00)00292-2CrossrefGoogle Scholar

  • [11] KIRYAKOVA, V.: The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus, Comput. Math. Appl. 59 (2010), 1885–1895. http://dx.doi.org/10.1016/j.camwa.2009.08.025CrossrefGoogle Scholar

  • [12] KIRYAKOVA, V.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl. 59 (2010), 1128–1141. http://dx.doi.org/10.1016/j.camwa.2009.05.014CrossrefGoogle Scholar

  • [13] MARKUSHEVICH, A.: A Theory of Analytic Functions 1, 2, Nauka, Moscow, 1967 (Russian). Google Scholar

  • [14] MATHAI, A. M.— HAUBOLD, H. J.: Special Functions for Applied Scientists, Springer, New York, 2008. http://dx.doi.org/10.1007/978-0-387-75894-7CrossrefGoogle Scholar

  • [15] PANEVA-KONOVSKA, J.: Cauchy-Hadamard, Abel and Tauber type theorems for series in generalized Bessel-Maitland functions, C. R. Acad. Bulgare Sci. 61 (2008), 9–14. Google Scholar

  • [16] PANEVA-KONOVSKA, J.: Theorems on the convergence of series in generalized Lommel-Wright functions, Fract. Calc. Appl. Anal. 10 (2007), 59–74. Google Scholar

  • [17] PANEVA-KONOVSKA, J.: Tauberian theorem of Littlewood type for series in Bessel functions of first kind, C. R. Acad. Bulgare Sci. 62 (2009), 161–166. Google Scholar

  • [18] PANEVA-KONOVSKA, J.: Convergence of series in some multi-index Mittag-Leffler functions. In: Proc. of 4th IFAC Workshop “Fractional Differentiation and its Applications’ 2010”, Badajoz, Spain, Oct. 18–20, 2010 (I. Podlubny, B. M. Vinagre Jara, YQ. Chen, V. Feliu Batlle, I. Tejado Balsera, eds.), 2010, Article No FDA10-147, pp. 1–4. Google Scholar

  • [19] PANEVA-KONOVSKA, J.: Convergence of series in Mittag-Leffler functions, C. R. Acad. Bulgare Sci. 63 (2010), 815–822. Google Scholar

  • [20] PANEVA-KONOVSKA, J.: Series in Mittag-Leffler functions: inequalities and convergent theorems, Fract. Calc. Appl. Anal. 13 (2010), 403–414. Google Scholar

  • [21] PANEVA-KONOVSKA, J.: Inequalities and asymptotic formulae for the three parametric Mittag-Leffler functions, Math. Balkanica (N.S.) 26 (2012), 203–210. Google Scholar

  • [22] PODLUBNY, I.: Fractional Differential Equations, Acad. Press, San Diego, 1999. Google Scholar

  • [23] PRABHAKAR, T. R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. Google Scholar

  • [24] RAJKOVIC, P.— MARINKOVIC, S.— STANKOVIC, M.: Diferencijalno-Integralni Racun Baicnih Hipergeometrijskih Funkcija, Mas. Fak. Univ. u Nisu, Nis, 2008 (Serbian). Google Scholar

  • [25] RUSEV, P.: A theorem of Tauber type for the summation by means of Laguerre’s polynomials, C. R. Acad. Bulgare Sci. 30 (1977), 331–334 (Russian). Google Scholar

  • [26] RUSEV, P.: Analytic Functions and Classical Orthogonal Polynomials. Bulgarian Math. Monographs, No. 3, Publ. House Bulg. Acad. Sci., Sofia, 1984. Google Scholar

  • [27] RUSEV, P.: Classical Orthogonal Polynomials and Their Associated Functions in Complex Domain. Bulgarian Acad. Monographs, No. 10, Marin Drinov Acad. Publ. House, Sofia, 2005. Google Scholar

  • [28] SANDEV, T.— TOMOVSKI, Ž.— DUBBELDAM, J.: Generalized Langevin equation with a three parameter Mittag-Leffler noise, Phys. A, 390 (2011), 3627–3636 http://dx.doi.org/10.1016/j.physa.2011.05.039CrossrefGoogle Scholar

About the article

Published Online: 2014-03-06

Published in Print: 2014-02-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 1, Pages 73–84, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-013-0188-0.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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