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

•Caputo fractional derivative of analytic functions (not necessarily of exponential order).•Commutative and semigroup properties of Caputo derivative, under less restrictive requirements.•Properties of Caputo-type convolution operator defined by Bernstein functions. 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 properties for a generalization of the above fractional derivative, i.e. the Caputo-type convolution 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.