Pathway Fractional Integral Operator Associated with 3m-Parametric Mittag-Leffler Functions

  • Shilpi Jain
  • Praveen Agarwal
  • Adem KilicmanEmail author
Original Paper


In this paper, we present composition of the pathway fractional integral \(P_{0^{+} }^{(\eta ,\alpha )}\) with the 3m-parametric type Mittag-Leffler function \(E^{(\gamma _{i}),m}_{(\alpha _i), (\beta _i)}(z)\) and discusses some of it’s particular cases in application point of view.


Pathway fractional integral operators Multi-index Mittag-Leffler function Generalized wright function 

Mathematics Subject Classification

26A33 33E12 Secondary 33C60 33E20 



This work has done during the visit of second author at Universiti Putra Malaysia. Thus the second and third authors are very greateful to University Putra Malaysia for the partial support under the reserach Grant having No. UPM-IPS 9543000.

Compliance with Ethical Standards

Conflict of interest

There is no conflict of interests.

Author Contributions

All authors contributed equally to the manuscript and approved the final manuscript.


  1. 1.
    Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 51 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kilbas, A.A., Koroleva, A.A., Rogosin, S.V.: Multi-parametric Mittag-Leffler functions and their extension. Fract. Calc. Appl. Anal. 16(2), 378–404 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform. Spec. Funct. 15, 31–49 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland Mathematics Studies 204, Amsterdam. New York (2006)CrossRefGoogle Scholar
  5. 5.
    Kiryakova, V.: Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal. 2(4), 445–462 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kiryakova, V.: Some special functions related to fractional calculus and fractional (non-integer) order control systems and equations. Facta Universitatis (Sci. J. Univ. Nis), Ser. Autom. Control Robot. 7(1), 79–98 (2008)MathSciNetGoogle Scholar
  7. 7.
    Kiryakova, V.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kiryakova, V.S.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59(3), 1128–1141 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kiryakova, V.S.: The multi-index Mittag-Leffler function as an important class of special functions of fractional calculus. Comput. Math. Appl. 59(5), 1885–1895 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kiryakova, V.S., Luchko, Y.F.: The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis. AIP Conf. Proc. 1301, 597 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mathai, A.M., Saxena, R.K.: The \(H\)-function with Applications in Statistics and Other Disciplines. Halsted Press, Sydney (1978)zbMATHGoogle Scholar
  12. 12.
    Nair, S.S.: Pathway fractional integration operator. Fract. Calc. Appl. Anal. 12(3), 237–252 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Paneva-Konovska, J.: On the multi-index (\(3m\)-parametric) Mittag-Leffler functions, fractional calculus relations and series convergences. Cent. Eur. J. Phy. 11(10), 1164–1177 (2013). CrossRefGoogle Scholar
  14. 14.
    Paneva-Konovska, J.: From Bessel to Multi-index Mittag-Leffler Functions: Enumerable Families, Series in Them and Convergence, London, World Scientific Publ. Singapore. (2016)
  15. 15.
    Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht (2001)CrossRefGoogle Scholar
  17. 17.
    Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)zbMATHGoogle Scholar
  18. 18.
    Srivastava, H.M., Gupta, K.C., Goyal, S.P.: The \(H\)-functions of One and Two Variables with Applications. South Asian, New Delhi (1982)zbMATHGoogle Scholar
  19. 19.
    Srivastava, H.M., Saxena, R.K.: Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Srivastava, H.M., Tomovski, Ž.: Fractional claculus with an integral operator containing generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211(1), 198–110 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wright, E.M.: The asymptotic expansion of the generalized Bessel function. Proc. Lond. Math. Soc. 38(2), 257–270 (1934). 50MathSciNetzbMATHGoogle Scholar
  22. 22.
    Wright, E.M.: The asymptotic expansion of the generalized hypergeometric function. J. Lond. Math. Soc. 10, 286–293 (1935)CrossRefGoogle Scholar
  23. 23.
    Yakubovich, S., Luchko, Y.: The Hypergeometric Approch to Integral Transforms and Convolutions, 1st edn. Kluwer Academic, Dordrecht (1994)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Department of MathematicsPoornima College of EngineeringJaipurIndia
  2. 2.Department of MathematicsAnand International College of EngineeringJaipurIndia
  3. 3.Department of MathematicsUniversity Putra MalaysiaSerdangMalaysia

Personalised recommendations