  Much effort has been devoted in the last two decades to characterize the
situations in which a reservoir computing system exhibits the so-called echo
state (ESP) and fading memory (FMP) properties. These important features
amount, in mathematical terms, to the existence and continuity of global
reservoir system solutions. That research is complemented in this paper with
the characterization of the differentiability of reservoir filters for very
general classes of discrete-time deterministic inputs. This constitutes a novel
strong contribution to the long line of research on the ESP and the FMP and, in
particular, links to existing research on the input-dependence of the ESP.
Differentiability has been shown in the literature to be a key feature in the
learning of attractors of chaotic dynamical systems. A Volterra-type series
representation for reservoir filters with semi-infinite discrete-time inputs is
constructed in the analytic case using Taylor's theorem and corresponding
approximation bounds are provided. Finally, it is shown as a corollary of these
results that any fading memory filter can be uniformly approximated by a finite
Volterra series with finite memory.
