A new Leibniz rule and its integral analogue for fractional derivatives

In 1972, T.J. Osler proposed a generalization of the Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function. This new rule was based on one of his own result on Taylor's series for the fractional derivative he obtained in 1971. Later, he gave the integral analogue of that new Leibniz rule. In this paper, we present a new Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function and we give its integral analogue. Finally, new series expansions and definite integrals involving special functions are derived as special cases of the new Leibniz rule and the corresponding integral analogue.