Commutative and associative properties of the Caputo fractional derivative and its generalizing convolution operator (Q2208134)

Whereas classical derivatives of integer order satisfy the semigroup property \(D^m D^n f = D^{m+n} f\) (\(m, n \in \mathbb N\)) for a very large and well understood class of functions \(f\), much less is known about such relations in fractional calculus, i.e., in the case where \(m,n \in (0, \infty) \setminus \mathbb N\). In fact, it is quite common to see publications where this relation is applied without any justification. The paper under review yields some progress in this context by providing a novel set of sufficient conditions on the function \(f\) under which the identity above is true when the fractional derivative is defined in the sense of Caputo or by a convolution structure that slightly generalizes the Caputo concept.