The convolution method for the development of new leibniz rules involving fractional derivatives and of their integral analogues

Leibniz rules for the operator of fractional derivatives and their integral analogues were considered, among other workers on the subject of fractional calculus, by T.J. Osler and Y. Watanabe. This operator of fractional derivatives is known to belong to a class of integral transforms associated with the Mellin convolution. Since Mejer' G-function transformation happens to be one of the most general integral transforms of this class, it would seem natural to obtain some new Leibniz rules and their integral analogues for operators involving the G-transforms. The object of the present paper is to summarize the development and applications of a general method for constructing such Leibniz rules and their integral analogues based upon the notion of the G-convolution and upon various representations of the kernels fo G-convolutions.