A Study of Marichev-Saigo-Maeda Fractional Integral Operators Associated with the S-Generalized Gauss Hypergeometric Function

  • Bansal, Manish Kumar (Department of Applied Science, Govt. Engineering College) ;
  • Kumar, Devendra (Department of Mathematics, University of Rajasthan) ;
  • Jain, Rashmi (Department of Mathematics, Malaviya National Institute of Technology)
  • Received : 2017.07.30
  • Accepted : 2018.12.20
  • Published : 2019.09.23


In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.


S-Generalized Gauss hypergeometric function;Marichev-Saigo-Maeda fractional integral operators;Appell's function;Mellin transform


  1. L. Debnath and D. Bhatta, Integral transforms and their applications, Chapman & Hall/CRC, Boca Raton, London, New York, 2007.
  2. A. Choudhary, D. Kumar and J. Singh, Numerical simulation of a fractional model of temperature distribution and heat flux in the semi infinite solid, Alexandria Eng. J., 55(1)(2016), 87-91.
  3. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies 204, Elsevier Science Publishers, Amsterdam, London and New York, 2006.
  4. D. Kumar, J. Singh and D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynam., 87(2017), 511-517.
  5. D. Kumar, J. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40(15)(2017), 5642-5653.
  6. D. Kumar, J. Singh and D. Baleanu, Modified Kawahara equation within a fractional derivative with non-singular kernel, Thermal Science, 22(2018), 789-796.
  7. O. I. Marichev, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izvestiya Akademii Nauk BSSR. Seriya Fiziko-Matematicheskikh Nauk, 1(1974), 128-129.
  8. A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-function: theory and applications, Springer, New York, 2010.
  9. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Willey & Sons, New York, NY, 1993.
  10. I. Podlubny, Fractional differential equations, Academic Press, San Diego, CA,1999.
  11. M. Saigo and N. Maeda, More generalization of fractional calculus, Transform Methods & Special Functions, Varna '96, 386-400, Bulgarian Acad. Sci., Sofia, 1998.
  12. J. Singh, D. Kumar and J. J. Nieto, A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow, Entropy, 18(6)(2016), Paper No. 206, 8 pp.
  13. J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, Analysis of a new fractional model for damped Burgers' equation, Open Physics, 15(2017), 35-41.
  14. H. M. Srivastava and P. Agarwal, Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math., 8(2)(2013), 333-345.
  15. H. M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 247(2014), 348-352.
  16. H. M. Srivastava , R. Jain and M. K. Bansal, A study of the S-generalized Gauss hypergeometric function and its associated integral transforms. Turkish J. Anal. Number Theory, 3(5)(2015), 116-119.
  17. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1985.
  18. H. M. Srivastava, D. Kumar and J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45(2017), 192-204.
  19. H. M. Srivastava and M. Saigo, Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation, J. Math. Anal. Appl., 121(2)(1987), 325-369.
  20. H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput., 118(2001), l-52.
  21. H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete  H-functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys., 25(1)(2018), 116-138.
  22. X.-J. Yang, D. Baleanu and H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam, 2016.