Differential calculus for fully matricial functions I

We will introduce a cyclic derivative for fully (stably) matricial functions and study its basic properties. In particular, we will show the Poincaré lemma for stably matricial functions of certain classes. We will also position Voiculescu’s framework of fully matricial functions in the context of nc functions due to Kaliuzhnyi-Verbovetskyi and Vinnikov in order to clarify the relation between the present work and previous related works.