Derivatives of the tree function

Abstract

We study some sequences of polynomials that appear when we consider the successive derivatives of the tree function (or Lambert's W function). We show in particular that they are related with a generalization of Cayley trees, called Greg trees. Besides the combinatorial result in itself, it is interesting to see how this is related with previous work: similar problems were considered first by Ramanujan, and more recently in the theory of completely monotonic functions and its link with probability. Also of great interest is the fact that these Greg trees were introduced in a problem of textual criticism, as a kind a genealogical trees, where they had a priori no mathematical meaning.