International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 5, Number 2 (2013), pp. 43-54 © International Research Publication House http://www.irphouse.com Saigo-Maeda Fractional Differential Operators of the Multivariable H-Function Kantesh Gupta and Meena Kumari Gurjar Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, Rajasthan (India) E-mail: kanteshgupta1@gmail.com, meenanetj@gmail.com Abstract In this paper, we study and develop the generalized fractional differential operators involving Appell’s function () . 3 F [1, p. 224, Eq. 5.7.1 (8)] introduced by Saigo and Maeda [2, p. 393] to the multivariable H-function. First, we establish two theorems that give the images of the multivariable H- function in Saigo-Maeda operators. On account of general nature of these operators, a large number of new and known theorems involving Saigo, Riemann-Liouville, Erdélyi- Kober fractional differential operators and several special functions notably generalized wright hypergeometric function, Mittag-Leffler function, generalized lauricella function, Bessel functions follow as special cases of our main findings. The important results obtained by Gupta [3], Kilbas [4], Kilbas and Saigo [5], Kilbas and Sebastian [6], Saxena, Ram and Suthar [7] and Saxena and Saigo [8] follow as special cases of our results. Keywords: Saigo-Maeda fractional differential operators, Appell function, Gauss hypergeometric function, Multivariable H-function, Bessel function, Mittag-Leffler function. AMS Subject Classification: 26A33, 33C05, 33C10, 33C60, 33C65, 33E12. 1. Introduction: The fractional differential operator involving various special functions, have been found significant importance and applications in various sub-field of application mathematical analysis. Since last five decades, a number of workers like Kiryakova [9], Srivastava et al. [10], Saxena et al. [11, 12], Saigo [13], Kilbas [4], Kilbas and