Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator

While for the integer-order derivatives, the commutative and semigroup (or associative) properties hold, the same is not true, in general, for the fractional derivatives. We show that, when the function is analytic, the Caputo derivative enjoys the above mentioned properties, at least when the fractional indices are smaller than one. This result is proved under assumptions on the derivatives (evaluated in zero) that are much less restrictive than the usual requirement of vanishing in the origin. Finally, we study the same prop- erties for a generalization of the above fractional derivative, i.e. the Caputo-type convolu- tion operator defined in [12] and [25]. In this more general setting (which includes the fractional, as particular case), we prove that the commutative property holds, while the associative property must be formulated accordingly.