Julia’s equation and differential transcendence

Abstract

We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially transcendental over the ring of convergent power series. Our results apply, in particular, to the exponential generating function of a sequence arising from work of Shadrin and Zvonkine on Hurwitz numbers.