International Journal of Scientific and Research Publications, Volume 5, Issue 1, January 2015 1 ISSN 2250-3153 www.ijsrp.org Double Dirichlet Average of M-series and Fractional Derivative Mohd. Farman Ali 1 , Renu Jain 2 , Manoj Sharma 3 1,2 School of Mathematics and Allied Sciences, Jiwaji University, Gwalior, Address 3 Department of Mathematics RJIT, BSF Academy, Tekanpur, Address Abstract- In this present paper we establish the results of Double Dirichlet average of M-Series and use fractional derivative. Some particular cases of our result are the generalization of earlier results. Index Terms- Dirichlet average, M-Series, fractional calculus operators. Mathematics Subject Classification: 26A33, 33A30, 33A25 and 83C99. I. INTRODUCTION arlson [1-5] has defined Dirichlet average of functions which represents certain type of integral average with respect to Dirichlet measure. He showed that various important special functions can be derived as Dirichlet averages for the ordinary simple functions like , etc. He has also pointed out [3] that the hidden symmetry of all special functions which provided their various transformations can be obtained by averaging , etc. Thus he established a unique process towards the unification of special functions by averaging a limited number of ordinary functions. Almost all known special functions and their well known properties have been derived by this process. Recently, Gupta and Agarwal [9, 10] found that averaging process is not altogether new but directly connected with the old theory of fractional derivative. Carlson overlooked this connection whereas he has applied fractional derivative in so many cases during his entire work. Deora and Banerji [6] have found the double Dirichlet average of e x by using fractional derivatives and they have also found the Triple Dirichlet Average of x t by using fractional derivatives [7]. In the present paper the Dirichlet average of M-Series has been obtained. II. DEFINITIONS Some of the definitions which are necessary in the preparation of this paper. 1.1 Standard Simplex in : Denote the standard simplex in , by [1, p.62]. 1.2 Dirichlet measure: Let and let be the standard simplex in The complex measure is defined by . knows as Dirichlet measure. Here Open right half plane and k is the Cartesian power of C