On the concept of derivatives and Taylor expansions for nonlinear input-output systems

Whenever the time interval is not fixed, i.e., when the dynamics is taken into account, Volterra series are not functional Taylor expansions as too often falsely asserted. It is shown here that with respect to a new "causal" derivative, which is related to the shuffle product, noncommutative generating power series are genuine Taylor expansions. An implicit function theorem is proved.