The extended Mittag-Leffler function via fractional calculus

Abstract

In this study, our main attempt is to introduce fractional calculus (fractional integral and differential) operators which contain the following new family of extended Mittag-Leffler function:

E-alpha,beta(gamma;q,c) (z) = Sigma(infinity)(n=0) B-p (gamma + nq, c - gamma)(c)(nq) z(n)/B(gamma, c - gamma)Gamma(alpha n + beta) n!' (z,beta,gamma is an element of C),

as its kernel. We also investigate a certain number of their consequences containing the said function in their kernels.