Fractional derivatives satisfy quite well all the properties that one could expect from them, despite some of them are only characteristic of integer order differentiation and some other have restrictions. For instance, the property of linearity (1.10) is fulfilled, while that of the iteration (1.9) has some restrictions in the cases that positive differentiation orders are present.
Some properties that include summatories can be generalized changing the summatories into integrals. One such property is the expansion in Taylor series
and other is the Leibniz rule
These and other generalized properties can be applied to the study of special functions, which often can be expressed in terms of simple formulas involving fractional derivatives. For instance, Gauss's hypergeometric function can be expressed as