Abstract.

Our main objective in this survey-cum-expository article is essentially to present a review of some recent developments involving various classes of the Mittag-Leffler type functions which are associated with several family of generalized Riemann-Liouville and other related fractional derivative operators. Specifically, we consider various compositional properties, which are associated with the Mittag-Leffler type functions and the Hardy-type inequalities for a certain generalized fractional derivative operator. We also present solutions of many different classes of fractional differential equations with constant coefficients and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions as well as a number of interesting particular cases of these general solutions and certain recently investigated fractional kinetic differintegral equations.

Keywords:

Riemann-Liouville, Liouville-Caputo fractional derivative operator, generalized MittagLeffler function, Volterra differintegral, Lebesgue integrable functions, Fox-Wright hypergeometric functions, Hurwitz-Lerch and related zeta functions.