Document Type: Research Paper

Author

Department of Mathematics & Statistics, Jai Narain Vyas University, Jodhpur - 342005, India.

Abstract

The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The Marichev-Saigo-Maeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide extensions of the results given by Gupta and Parihar [3], Saxena and Saigo [30], Samko et al. [26]. On account of the general nature of the generalized Mittag-Leffler function and generalized Wright function, a number of known results can be easily found as special cases of our main results.

Keywords

[1] J. Choi and D. Kumar, Certain uni ed fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, Journal of Inequalities and Applications, 2014 (2014), 15 pages.

[2] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Tables of Integral Transforms, McGraw-Hill, New York, 1, 1954.

[3] A. Gupta and C.L. Parihar, Fractional di erintegral operators of the generalized Mittag-Leer function, Bol. Soc. Paran. Math., 33(1) (2015), 137-144.

[4] H.J. Haubold, A.M. Mathai, and R.K. Saxena, Mittag-Leer functions and their applications, J. Appl. Math. (Article ID 298628) (2011), 1-51.

[5] A.A. Kilbas and M. Saigo, Fractional integrals and derivatives of Mittag-Leer type function, Doklady Akad. Nauk Belarusi, 39(4) (1995), 22-26.

[6] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leer function and generalized fractional calculus operators, Integral Transform Special Function, 15 (2004), 31-49.

[7] A.A. Kilbas, M. Saigo and J.J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal., 5(4) (2002), 437460.

[8] Y.C. Kim, K.S. Lee and H.M. Srivastava, Some applications of fractional integral operators and Ruscheweyh derivatives, J. Math. And. Appl., 197(2) (1996), 505-517.

[9] V. Kiryakova, All the special functions are fractional di erintegrals of elementary functions, Journal of Physics A: Mathematical and General, 30(14) (1997), 5085-5103.

[10] D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 14(3) (2014), 25-36.

[11] D. Kumar and S. Kumar, Fractional Calculus of the Generalized Mittag-Leer Type Function, International Scholarly Research Notices 2014 (2014), Article ID 907432, 6 pages.

[12] D. Kumar and S.D. Purohit, Fractional di erintegral operators of the generalized Mittag-Leer type function, Malaya J. Mat., 2(4) (2014), 419-425.

[13] D. Kumar and R.K. Saxena, Generalized fractional calculus of the M-Series involving F3 hypergeometric function, Sohag J. Math., 2(1) (2015), 17-22.

[14] 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, (Russian).

[15] A.M. Mathai and H.J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008.

[16] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Di erential Equations, John Wiley & Sons, New York, NY, USA, 1993.

[17] G.M. Mittag-Leer, Sur la nouvelle fonction E (x), C.R. Acad. Sci. Paris 137 (1903), 554-558.

[18] G.M. Mittag-Leer, Sur la representation analytique d'une branche uniforme d'une function monogene, Acta Math. 29 (1905), 101-181.

[19] J. Paneva-Konovska, Inequalities and asymptotic formulae for the three para-metric Mittag-Leer functions, Math. Balkanica, 26 (2012), 203-210.

[20] J. Paneva-Konovska, Convergence of series in three parametric Mittag-Leer functions, Math. Slovaca 62, 2012.

[21] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leer function in the Kernel, Yokohama Math. J. 19 (1971), 7-15.

[22] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep., College General Ed. Kyushu Univ. 11 (1978), 135-143.

[23] M. Saigo and N. Maeda, More generalization of fractional calculus Transform Methods and Special Functions, Varna, Bulgaria, (1996), 386-400.

[24] T.O. Salim, Some properties relating to the generalized Mittag-Leer function, Adv. Appl. Math. Anal., 4 (2009), 21-30.

[25] T.O. Salim and A.W. Faraj, A generalization of Mittag-Leer function and Integral operator associated with fractional calculus, Journal of Fractional Calculus and Application, 3(5) (2012), 1-13.

[26] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon et alibi, 1993.

[27] R.K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph-function involving a general class of polynomials, Acta Mathematica Scientia, 35(5) (2015), 1095-1110.

[28] R.K. Saxena, J. Ram and D. Kumar, Generalized fractional di erentiation of the Aleph-Function associated with the Appell function F3 , Acta Ciencia Indica, 38M(4) (2012), 781-792.

[29] R.K. Saxena, J. Ram and D. Kumar, On the Two-Dimensional Saigo-Maeda fractional calculus associated with Two-Dimensional Aleph Transform, Le Matematiche, 68 (2013), 267-281.

[30] R.K. Saxena and M. Saigo, Certain properties of the fractional calculus operators associated with generalized Mittag-Leer function, Fract. Calc. Appl. Anal., 8(2) (2005), 141-154.

[31] H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leer function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.

[32] A. Wiman, Uber de fundamental satz in der theorie der funktionen E (x), Acta Math. 29 (1905), 191-201.

[33] E.M. Wright, The asymptotic expansion of generalized hypergeometric function, J. London Math. Soc., 10 (1935), 286-293.