On the fractional calculus of generalized Mittag-Leffler function

Abstract

The paper is devoted to the study of generalized fractional calculus of the generalized Mittag-Leffler function \(E^{\delta}_{\nu,\rho}(z)\) which is an entire function of the form \[E^{\delta}_{\nu,\rho}(z) = \sum_{s=0}^{\infty}\frac{(\delta)_s z^s}{\Gamma(\nu n + \rho)s!}\], where \(\nu > 0\) and \(\rho > 0\). For \(\delta = 1\), it reduces to Mittag-Leffler function \(E_{\nu,\rho}(z).\) We have shown that the generalized functional calculus operators transform such functions with power multipliers in to generalized Wright function. Some elegant results obtained by Kilbas and Saigo [11], Saxena and Saigo [24] are the special cases of the results derived in this paper.