Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
See all formats and pricing
More options …
Volume 13, Issue 1


Volume 13 (2015)

Some fractional integral formulas for the Mittag-Leffler type function with four parameters

Praveen Agarwal / Juan J. Nieto
  • Departamento de Análise Matemática, Facultade de Matemáicas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain and Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-09-25 | DOI: https://doi.org/10.1515/math-2015-0051


In this paper we present some results from the theory of fractional integration operators (of Marichev- Saigo-Maeda type) involving the Mittag-Leffler type function with four parameters ζ , γ, Eμ, ν[z] which has been recently introduced by Garg et al. Some interesting special cases are given to fractional integration operators involving some Special functions.

Keywords: Marichev-Saigo-Maeda type fractional integral operators; Mittag-Leffler type function with four parameters; Generalized Wright function


  • [1] Agarwal P, Trujillo J J, Rogosin S V, Certain fractional integral operators and the generalized multiindex Mittag-Leffler functions, Proc. Indian Acad. Sci. Math. Sci. (In press)Google Scholar

  • [2] Agarwal P, Chnad M and Jain S, Certain integrals involving generalized Mittag-Leffler functions, Proc. Nat. Acad. Sci. India Sect. A, (2015); doi:10.1007/s40010-015-0209-1CrossrefGoogle Scholar

  • [3] Agarwal P, Jain S, Chnad M, Dwivedi S K, Kumar S, Bessel functions Associated with Saigo-Maeda fractional derivative operators, J. Fract. Calc. 5(2) (2014) 102-112Google Scholar

  • [4] Al-Bassam M A, Luchko Y F, On generalized fractional calculus and it application to the solution of integro-differential equations. J. Fract. Calc. 7 (1995) 69-88Google Scholar

  • [5] Baleanu D, Diethelm K, Scalas E, Trujillo J J, Fractional Calculus: Models and Numerical Methods (2012) (N. Jersey, London, Singapore: World Scientific Publishers)Google Scholar

  • [6] Capelas de Oliveira E, Mainardi F, VAZ J Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J. Special Topics 193 (2011) 161-171;http://dx.doi.org/10.1140/epjst/e2011-01388-0 CrossrefGoogle Scholar

  • [7] Caponetto R, Dongola G, Fortuna L, and Petráš I, Fractional Order Systems: Modeling and Control Applications (2010) (Singapore: World Scientific Pub Co Inc)Google Scholar

  • [8] Caputo M, Mainardi F, Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II). 1 (1971) 161-198Google Scholar

  • [9] Choi J, Agarwal P, A note on fractional integral operator associate with multiindex Mittag-Leffler functions, Filomat (In press)Google Scholar

  • [10] Choi J, Agarwal P, Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions, Abstr. Appl. Anal. 2014 (2014) 735946,7 pages;available online at http://dx.doi.org/10.1155/2014/735946 CrossrefGoogle Scholar

  • [11] Diethelm K, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type (2010) (Berlin: Springer) Springer Lecture Notes in Mathematics No 2004Google Scholar

  • [12] Dzrbashjan M M, On the integral transforms generated by the generalized Mittag-Leffler function, Izv. AN Arm. SSR 13(3) (1960) 21-63Google Scholar

  • [13] Garg M, A. Sharma and P. Manohar, A Generalized Mittag-Leffler Type Function with Four Parameters, Thai.J. Math., (In press)Google Scholar

  • [14] Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer 2014) 454 pages. http://www.springer.com/us/book/9783662439296 Google Scholar

  • [15] Gorenflo R, Mainardi F, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri and F. Mainardi (Editors) Fractals and Fractional Calculus in Continuum Mechanics 223-276 (1997) (Springer Verlag, Wien)Google Scholar

  • [16] Haubold H J, Mathai A M, and Saxena R K, Mittag-Leffler functions and their applications, J. Appl. Math. 2011 (2011) 298628, 51 pages;available online at http://dx.doi.org/10.1155/2011/298628 CrossrefGoogle Scholar

  • [17] Hilfer R (Edt), Applications of Fractional Calculus in Physics (2000) (New Jersey, London, Hong Kong:Word Scientific Publishing Co.)Google Scholar

  • [18] Kilbas A A, Koroleva A A, Rogosin S V, Multi-parametric Mittag-Leffler functions and their extension, Fract. Calc. Appl. Anal. 16(2) (2013) 378-404Google Scholar

  • [19] Kilbas A A, Saigo M and Saxena R K, Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels, J. Integral Equations Appl. 14(4) (2002) 377-386Google Scholar

  • [20] Kilbas A A, Srivastava H M, Trujillo J J, Theory and Applications of Fractional Differential Equations (2006)North-Holland Mathematics Studies. 204 (Elsevier, Amsterdam, etc)Google Scholar

  • [21] Kiryakova V,Generalized Fractional Calculus and Applications (1994) (Harlow, Longman)Google Scholar

  • [22] Kiryakova V, Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms, Fract. Calc. Appl. Anal. 2(4) (1999) 445-462Google Scholar

  • [23] Kiryakova V, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000) 241-259CrossrefGoogle Scholar

  • [24] Kiryakova V, On two Saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9(2) (2006) 160-176Google Scholar

  • [25] Kiryakova V S, The special functions of fractional calculus as generalized fractional calculus operators od some basic functions, Comp. Math. Appl. 59(3) (2010) 1128-1141Google Scholar

  • [26] Kiryakova V S, The multi-index Mittag-Leffler function as an important class of special functions of fractional calculus, Comp. Math. Appl. 59(5) (2010) 1885-1895Google Scholar

  • [27] Mainardi F, Fractional Calculus and Waves in Linear Viscoelasticity (2010) (London: Imperial College Press)Google Scholar

  • [28] Marichev O I, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izv. AN BSSR Ser. Fiz.-Mat. Nauk No. 1 (1974) 128-129Google Scholar

  • [29] Mathai A M, Saxena R K, The H-function with Applications in Statistics and Other Disciplines, Halsted Press [John Wiley & Sons], New York, London, Sydney, 1978Google Scholar

  • [30] Mathai A M, Saxena R K, Haubold H J, The H-function. Theory and Applications (2010) (Dordrecht: Springer)Google Scholar

  • [31] McBride A C, Fractional Calculus and Integral Transforms of Gen- eralized Functions (1979) (Research Notes in Math. 31) (Pitman, London)Google Scholar

  • [32] Miller K S, Ross B, An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) (New York: John Wiley and Sons)Google Scholar

  • [33] Mittag-Leffler G M, Sur la nouvelle fonction E .x/, C. R. Acad. Sci. Paris 137 (1903) 554-558Google Scholar

  • [34] NIST Handbook of Mathematical Functions. Edited by Frank W.J. Olver (editor-in-chief), D.W. Lozier, R.F. Boisvert, and C.W. Clark. Gaithersburg, Maryland, National Institute of Standards and Technology, and New York, Cambridge University Press, 951 + xv pages and a CD, (2010)Google Scholar

  • [35] Oldham K B, Spanier J, The Fractional Calculus (1974) (New York:Academic Press)Google Scholar

  • [36] Podlubny I, Fractional Differential Equations (1999) (New York: Academic Press)Google Scholar

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

  • [38] Prakasa Rao B L S, Statistical inference for fractional diffusion processes (2010) (Chichester: John Wiley & Sons Ltd.)Google Scholar

  • [39] Rabotnov Yu N, Elements of Hereditary Solid Mechanics (1980) (Moscow:MIR)Google Scholar

  • [40] Rogosin S.V., "The Role of the Mittag-Leffler Function in Fractional Modeling" Mathematics 2015, 3, 368-381; doi:10.3390/math3020368CrossrefGoogle Scholar

  • [41] Saigo M, On generalized fractional calculus operators. In: Recent Advances in Applied Mathematics (Proc. Internat. Workshop held at Kuwait Univ.). Kuwait Univ., Kuwait, (1996) 441-450Google Scholar

  • [42] Saigo M, Maeda N, More generalization of fractional calculus, In: Transform Methods and Special Functions, Varna 1996 (Proc. 2nd Intern. Workshop, P. Rusev, I. Dimovski, V. Kiryakova Eds.), IMI-BAS, Sofia, (1998) 386-400Google Scholar

  • [43] Samko S G, Kilbas A A, Marichev O I, Fractional Integrals and Derivatives: Theory and Applications (1993) (New York and London: Gordon and Breach Science Publishers)Google Scholar

  • [44] Saxena R K and Nishimoto K, N-Fractional Calculus of Generalized Mittag- Leffler functions, J. Fract. Calc. 37(2010) 43-52Google Scholar

  • [45] Saxena R K, Saigo M, Generalized fractional calculus of the H-function associated with the Appell function, J. Fract. Calc. 19 (2001) 89-104 Google Scholar

  • [46] Shukla A K and Prajapati J C, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007) 797-811Google Scholar

  • [47] Srivastava H M and Agarwal P, Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math. 8(2) (2013) 333-345Google Scholar

  • [48] Srivastava H M, Choi J, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001Google Scholar

  • [49] Srivastava H M, Choi J, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012Google Scholar

  • [50] Srivastava H M, Gupta K C, Goyal S P, The H-functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, Madras, 1982Google Scholar

  • [51] Srivastava H M and Saigo M, Multiplication of fractional calculus operators and boundary value problems involving the eulerdarboux equation, J. Math. Anal. Appl. 121 (1987) 325-369CrossrefGoogle Scholar

  • [52] Srivastava H M, Tomovski LZ, Fractional claculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211(1) (2009) 198-210CrossrefGoogle Scholar

  • [53] Tarasov V E, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (2010) (Beijing: Springer, Heidelberg; Higher Education Press)Google Scholar

  • [54] Tenreiro Machado J A, Kiryakova V, Mainardi F, A poster about the old history of fractional calculus, Fract. Calc. Appl. Anal. 13 (4)(2010) 447-454Google Scholar

  • [55] Tenreiro Machado J A, Kiryakova V, Mainardi F, A poster about the recent history of fractional calculus, Fract. Calc. Appl. Anal. 13(3)(2010) 329-334Google Scholar

  • [56] Tenreiro Machado J A, Kiryakova V, Mainardi F, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat. 16(3) (2011) 1140-1153; available online at http://dx.doi.org/10.1016/j.cnsns.2010.05.027 CrossrefGoogle Scholar

  • [57] Uchaikin V V, Fractional derivatives for physicists and engineers. Vol. I. Background and theory(2013) (Beijing: Springer, Berlin - Higher Education Press)Google Scholar

  • [58] Uchaikin V V, Fractional derivatives for physicists and engineers, Vol. II Applications(2013) (Beijing: Springer, Berlin - Higher Education Press)Google Scholar

  • [59] Wiman A, Über den Fundamentalsatz in der Theorie der Funktionen E .x/, Acta Math. 29(1905) 191-201Google Scholar

  • [60] Zaslavsky G M, Hamiltonian Chaos and Fractional Dynamics (2005) (Oxford: Oxford University Press)Google Scholar

About the article

Received: 2015-06-16

Accepted: 2015-08-16

Published Online: 2015-09-25

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0051.

Export Citation

© 2015. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Chung-Sik Sin, Gang-Il Ri, and Mun-Chol Kim
Advances in Difference Equations, 2017, Volume 2017, Number 1
Gauhar Rahman, Praveen Agarwal, Shahid Mubeen, and Muhammad Arshad
Boletín de la Sociedad Matemática Mexicana, 2017
K. S. Nisar, S. R. Mondal, and P. Agarwal
Mathematical Problems in Engineering, 2016, Volume 2016, Page 1

Comments (0)

Please log in or register to comment.
Log in