Fractional calculus operators and their applications involving power functions and summation of series

Abstract

Many earlier works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, et cetera). The main object of the present paper is to examine rather systematically (and extensively) some of the most recent contributions on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series. Various other classes of infinite sums found in the mathematical literature by these (or other) means, and their validity or hitherto unnoticed connections with some known results, are also considered.