Some Fractional Calculus Results Pertaining To Mittag-Leffler Type Functions

Open access


In this paper, we study the generalized fractional operators pertaining to the generalized Mittag-Leffler function and multi-index Mittag-Leffler function. Some applications of the established results associated with generalized Wright function are also deduced as corollaries. The results are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

[1] KILBAS, A.A., SRIVASTAVA, H.M., AND TRUJILLO, J.J. 2006. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 204.

[2] CATTANI, C., SRIVASTAVA H.M. AND YANG X.J. 2015. Fractional Dynamics. Emerging Science Publishers (De Gruyter Open), Berlin and Warsaw.

[3] YANG, X.J., BALEANU, D. AND SRIVASTAVA, H.M. 2016. Local Fractional Integral Transforms and Their Applications. Academic Press (Elsevier Science Publishers), Amsterdam, Heidelberg, London and New York.

[4] SRIVASTAVA, H.M. 2016. Some families of Mittag-Leffler type functions and associated operators of fractional calculus. TWMS J. Pure Appl. Math., 7(2), 123-145.

[5] SRIVASTAVA, H.M. AND SAXENA, R.K. 2001. Operators and fractional integration and their applications. Appl. Math. Comput., 118, 1-52.

[6] SRIVASTAVA, H.M., KUMAR, D. AND SINGH, J. 2017. An efficient analytical technique for fractional model of vibration equation. Applied Mathematical Modelling 45, 192-204.

[7] MITTAG-LEFFLER, G.M. 1903. Sur la nouvelle fonction Ea(x), C.R. Acad. Sci., Paris (Ser.II), 137, 554-558.

[8] MITTAG-LEFFLER, G.M. 1905. Sur la representation analytique d’une branche uniforme d’une fonction monogene. Acta Math., 29, 101-181.

[9] PRABHAKAR, T.K. 1971. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J., 19, 7-15.

[10] SHUKLA, A.K. AND PRAJAPATI, J.C. 2007. On a generalized Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336, 797-811.

[11] SAXENA, R.K., RAM, J. AND VISHNOI, M. 2010. Fractional differentiation and fractional integration of the generalized Mittag-Leffler function. J. Indian Acad. Math., 32(1), 153-162.

[12] SAXENA, R.K. AND NISHIMOTO, K. 2011. Further results on the generalized Mittag-Leffler functions of fractional calculus. J. Fract. Calc., 40, 29-41.

[13] WRIGHT, E.M. 1935. The asymptotic expansion of the generalized hypergeometric functions. J. London Math. Soc. 10, 286-293.

[14] WRIGHT, E.M. 1934. The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc., 38(2), 257-270.

[15] KHAN, M.A. AND AHMED, S. 2013. On some properties of the generalized Mittag-Leffler function. Springer Plus a Springer Open Journal, doi:

[16] SAMKO, S.G., KILBAS, A.A. AND MARICHEV, O.I. 1993. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland.

[17] SAIGO, M. 1978. A Remark on integral operators involving the Gauss hypergeometric functions. Math .Rep. Kyushu Univ., 11, 135-143.

[18] SRIVASTAVA, H.M. AND KARLSSON, P.W. 1985. Multiple Gaussian hypergeometric series. Ellis Horwood, Chichester [John Wiley and Sons], New York.

[19] AHMED, S. 2014. On the generalized fractional integrals of the generalized Mittag-Leffler function. Springer Plus a Springer Open Journal, 3(1), 198.

[20] CHAURASIA, V.B.L. AND PANDEY, S.C. 2010. On the fractional calculus of the generalized Mittag-Leffler function. Scientia, Ser. A, Math. Sci., 20, 113-122.

Journal of Applied Mathematics, Statistics and Informatics

The Journal of University of Saint Cyril and Metodius

Journal Information

Mathematical Citation Quotient (MCQ) 2017: 0.06

Target Group

researchers in the fields of informatics, information technologies, statistics and mathematics


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 251 240 18
PDF Downloads 112 109 8