PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2013 | 11 | 10 | 1164-1177
Tytuł artykułu

On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we consider a family of 3m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright pΨq-functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.
Wydawca

Czasopismo
Rocznik
Tom
11
Numer
10
Strony
1164-1177
Opis fizyczny
Daty
wydano
2013-10-01
online
2013-12-19
Bibliografia
  • [1] A. Erdélyi et al. (Ed-s), Higher Transcendental Functions, Vols. 1–3, 1st edition (McGraw-Hill, New York-Toronto-London, 1953–1955)
  • [2] M. M. Dzrbashjan, Integral Transforms and Representations in the Complex Domain, 1st edition (Nauka, Moscow, 1966) (in Russian)
  • [3] I. Podlubny, Fractional Differential Equations, 1st edition (Acad. Press., Boston etc., 1999)
  • [4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 1st edition (Elsevier, Amsterdam etc., 2006)
  • [5] A. M. Mathai, H. J. Haubold, Special Functions for Applied Scientists, 1st edition (Springer, New York, 2008) http://dx.doi.org/10.1007/978-0-387-75894-7[Crossref]
  • [6] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 1st edition (Imperial College Press & World Sci., London - Singapore, 2010) http://dx.doi.org/10.1142/9781848163300[Crossref]
  • [7] V. Kiryakova, J. Comput. Appl. Math. 118, 241 (2000) http://dx.doi.org/10.1016/S0377-0427(00)00292-2[Crossref]
  • [8] V. Kiryakova, Comput. Math. Appl. 59, 1885 (2010) http://dx.doi.org/10.1016/j.camwa.2009.08.025[Crossref]
  • [9] V. Kiryakova, Comput. Math. Appl. 59, 1128 (2010) http://dx.doi.org/10.1016/j.camwa.2009.05.014[Crossref]
  • [10] R. R. Nigmatullin, D. Baleanu, Fract. Calc. Appl. Anal. 15, 718 (2012)
  • [11] R. Gorenflo, F. Mainardi, J. Comput. Appl. Math. 118, 283 (2000) http://dx.doi.org/10.1016/S0377-0427(00)00294-6[Crossref]
  • [12] V. Kiryakova, In: AIP Conf. Proc. 1410 (AMEE’2011), 247 (2011)
  • [13] M. A. E. Herzallah, D. Baleanu, Comput. Math. Appl. 64, 3059 (2012) http://dx.doi.org/10.1016/j.camwa.2011.12.060[Crossref]
  • [14] A. M. Mathai, H. J. Haubold, Fract. Calc. Appl. Anal. 14, 138 (2011)
  • [15] T. R. Prabhakar, Yokohama. Math. J. 19, 7 (1971)
  • [16] A. A. Kilbas, M. Saigo, R. K. Saxena, Integral Transforms Spec. Func. 15, 31 (2004) http://dx.doi.org/10.1080/10652460310001600717[Crossref]
  • [17] T. Sandev, R. Metzler, Z. Tomovski, Fract. Calc. Appl. Anal. 15 426 (2012)
  • [18] S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, 1st edition (Kluwer Acad. Publ., Dordrecht - Boston - London, 1994) http://dx.doi.org/10.1007/978-94-011-1196-6[Crossref]
  • [19] Yu. Luchko, Fract. Calc. Appl. Anal. 2, 463 (1999)
  • [20] V. Kiryakova, Fract. Calc. Appl. Anal. 2, 445 (1999)
  • [21] V. Kiryakova, Yu. Luchko, In: American Institute of Physics - Conf. Proc. 1301 (AMiTaNS’10), 597 (2010)
  • [22] A. A. Kilbas, A. A. Koroleva, S. V. Rogosin, Fract. Calc. Appl. Anal. 16, 378 (2013)
  • [23] J. Paneva-Konovska, C. R. Acad. Bulg. Sci. 61, 9 (2008)
  • [24] J. Paneva-Konovska, Fract. Calc. Appl. Anal. 10, 59 (2007)
  • [25] J. Paneva-Konovska, C. R. Acad. Bulg. Sci. 62, 75 (2009)
  • [26] J. Paneva-Konovska, C. R. Acad. Bulg. Sci. 63, 815 (2010)
  • [27] J. Paneva-Konovska, Integral Transforms Spec. Funct. 23, 207 (2012) http://dx.doi.org/10.1080/10652469.2011.575567[Crossref]
  • [28] J. Paneva-Konovska, C. R. Acad. Bulg. Sci. 64, 1089 (2011)
  • [29] T. Sandev, Ž. Tomovski, J. Dubbeldam, Physica A 390, 3627 (2011) http://dx.doi.org/10.1016/j.physa.2011.05.039[Crossref]
  • [30] J. Paneva-Konovska, Math. Balkanica (N.S.) 26, 203 (2012)
  • [31] J. Paneva-Konovska, Math. Slovaca (Accepted 12.01.2012)
  • [32] V. Kiryakova, Generalized Fractional Calculus and Applications, 1st edition (Longman & J. Wiley, Harlow - N. York, 1994)
  • [33] A. A. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. More Special Functions, 1st edition (Gordon & Breach Sci. Publ., N. York etc., 1990)
  • [34] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. 3rd edition (Chelsea Publ., New York, 1986) (the first edition Oxford Univ. Press, Oxford, 1937)
  • [35] Yu. Luchko, R. Gorenflo, Fract. Calc. Appl. Anal. 1, 63 (1998)
  • [36] M. M. Dzrbashjan, Izv. AN Arm. SSR 13, No 3, 21 (1960) (in Russian)
  • [37] E. M. Wright, J. London Math. Soc. 8, 71 (1933) http://dx.doi.org/10.1112/jlms/s1-8.1.71[Crossref]
  • [38] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, 1st edition (Gordon and Breach, N. York - London, 1993)
  • [39] G. Pagnini, Fract. Calc. Appl. Anal. 15, 117 (2012)
  • [40] P. Rusev, Classical Orthogonal Polynomials and Their Associated Functions in Complex Domain, 1st edition (Publ. House Bulg. Acad. Sci., Sofia, 2005)
  • [41] J. Paneva-Konovska, Adv. Math. Sci. J. 1, 73 (2012)
  • [42] G. Hardy, Divergent Series, 1st edition (Oxford University Press, Oxford, 1949)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_2478_s11534-013-0263-8
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.