Commutativity of the Leibniz rules in fractional calculus

Many earlier works on the subject of fractional calculus (that is, differentiation and integration of an arbitrary real or complex order) provide interesting accounts of the theory and applications of fractional calculus operators in several areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, etc.). The main object of this sequel to the aforementioned works is to examine rather closely the commutativity of the familiar Leibniz rules for fractional calculus and its various consequences. Some generalizations of a recent result of Tu, Chyan and Wu [1], involving fractional integration of powers of the logarithmic functions, are also considered.