A Note on \(K_4\) Fractional Integral Operator

  • Arif M. Khan
  • Pankaj Ramani
  • Daya Lal Suthar
  • Dinesh Kumar
Original Paper


The present paper deals with the study of new generalized fractional integral operator involving \(K_4\)-function due to Sharma. Mellin and Laplace transforms of this new operator are investigated. The bounded-ness and composition properties of the proposed operator are also established. Further, derived results are applied to solve fractional differential equation involving \(K_4\)-function associated with Hilfer derivatives. The \(K_4\)-function is further extension of M-series and the importance of desired results lies in the fact that many known results are readily follows as special cases of our finding. \(K_4\) and M-series have recently found essential application in solving problems of science, engineering and technology. Some special cases of the established results are given in form of corollaries.


\(K_4\)-function M-series Fractional integral operator H-function Generalized Wright function 

Mathematics Subject Classification

Primary 44A10 26A33 Secondary 33C20 33C05 33E12 



The author (Dinesh Kumar) would like to express his deep thanks to NBHM (National Board of Higher Mathematics), India, for granting a Post-Doctoral Fellowship (Sanction No. 2/40(37)/2014/R&D-II/14131).


  1. 1.
    Braaksma, B.L.J.: Asymptotic expansions and analytic continuations for a class of Barnes integrals. Compositio Math. 15, 239–341 (1964)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Faraj, A., Salim, T., Sadek, S., Ismail, J.: A generalization of M series and integral operator associated with fractional calculus. Asian J. Fuzzy Appl. Math. 2, 142–155 (2014)Google Scholar
  3. 3.
    Fox, C.: The \(G\) and \(H\)-functions as symmetrical Fourier kernels. Trans. Am. Math. Soc. 98, 395–429 (1961)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gupta, R.K., Shaktawat, B.S., Kumar, D.: A study of Saigo-Maeda fractional calculus operators associated with the Multiparameter \(K\)-Mittag-Leffler function. Asian J. Math. Comput. Res. 12(4), 243–251 (2016)Google Scholar
  5. 5.
    Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kalla, S.L.: Integral operators involving Fox’s \(H\)-function. Acta Mex. Cienc. Tecn. 3, 117–122 (1969)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Khan, A.M.: A note on Matichev-Saigo Maeda fractional integral operator. J. Fract. Cal. Appl. 5, 88–95 (2014)MathSciNetGoogle Scholar
  8. 8.
    Khan, A.M.: On certain new Cauchy type fractional integral inequalities and opial type fractional derivative inequalities. Tamkang J. Math. 46, 67–73 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Khan, A.M.: Generalized fractional integral operators and \(M\)-series. J. Math. 2016, 10 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kilbas, A.A., Saigo, M.: \(H\)-Transforms: Theory and Applications. Chapman and Hall/CRC, Boca Raton (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kiryakova, V.: On two Saigo’s fractional integral operators in the class of univalent functions. Frac. Calc. Appl. Anal. 9, 160–176 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kumar, D.: On certain fractional calculus operators involving generalized Mittag-Leffler function. Sahand Commun. Math. Anal. 3(2), 33–45 (2016)zbMATHGoogle Scholar
  13. 13.
    Kumar, D., Kumar, S.: Fractional integrals and derivatives of the generalized Mittag-Leffler type function. Int. Sch. Res. Not. 2014(907432), 5 (2014)Google Scholar
  14. 14.
    Kumar, D., Saxena, R.K.: Generalized fractional calculus of the \(M\)- series involving \(F_3\) hypergeometric function. Sohag J. Math. 2(1), 17–22 (2015)Google Scholar
  15. 15.
    Mathai, A.M., Saxena, R.K., Haubold, H.J.: The \(H\) Function Theory and Applications. Springer, New York (2010)zbMATHGoogle Scholar
  16. 16.
    McBride, A.C.: Fractional powers of a class of ordinary differential operators. Proc. Lond. Math. Soc. 45, 519–546 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Purohit, S.D., Kalla, S.L., Suthar, D.L.: Fractional integral operators and the multiindex Mittag-Leffler functions. Sci. Ser. A Math. Sci. (N.S.) 21, 87–96 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ram, J., Kumar, D.: Generalized fractional integration involving Appell hypergeometric of the product of two \(H\)-functions. Vijanana Parishad Anusandhan Patr. 54(3), 33–43 (2011)MathSciNetGoogle Scholar
  20. 20.
    Saxena, R.K., Chauhan, J.P., Jana, R.K., Shukla, A.K.: Further results on the generalized Mittag-Leffler function operator. J. Inequal. Appl. (2015). MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sharma, K.: On application of fractional differ-integral Operator to the \(K_4\)-function. Bol. Soc. Paran. Math. 30, 91–97 (2012)Google Scholar
  22. 22.
    Sharma, M., Jain, R.: A note on a generalized \(M\)-series as a special function of fractional calculus Frac. Calc. Appl. Anal. 12, 449–452 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Shukla, A., Prajapati, J.: On a generalization of Mittag-Leffler function and its properties. J. Math. Appl. Anal. 336, 797–811 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Srivastava, H.M., Tomovski, Z.: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the Kernel. Appl. Math. Comput. 211, 198–210 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Wright, E.M.: The asymptotic expansion of the Generalized hypergeometric functions. J. London Math. Soc. 10, 286–293 (1935)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsJodhpur Institute of Engineering and TechnologyJodhpurIndia
  2. 2.Department of MathematicsPoornima UniversityJaipurIndia
  3. 3.Department of MathematicsWollo UniversityWollo DessieEthiopia
  4. 4.Department of Mathematics & StatisticsJai Narain Vyas UniversityJodhpurIndia

Personalised recommendations