Julia's equation and differential transcendence (Q281483)

Let \(f\) be a non-linear entire function of the form NEWLINE\[NEWLINEf(z)=z+\sum_{k=p}^{\infty}f_kz^k \quad (p\geq 2,\; f_k \in \mathbb{C} \text{ for } k\geq p,\; f_p\neq 0).NEWLINE\]NEWLINE A way to describe the iteration of \(f\) in a full neighborhood of \(0\) is based on the functional equation \(\phi(f(z))=f'(z)\phi(z)\). A unique formal power series solution of the equation NEWLINE\[NEWLINE\phi(z)=f_pz^p+\sum_{k=p+1}^{\infty}\phi_kz^k \quad (\phi_k \in \mathbb{C} \text{ for } k\geq p+1)NEWLINE\]NEWLINE is called the \textit{iterative logarithm} of \(f\) and denoted by \(\text{itlog}(f).\)NEWLINENEWLINEThe main result of the paper is the statement that \(\text{itlog}(f)\) is \textit{differentially transcendental} over the ring of entire functions, that is \(\phi\) does not satisfy a non-trivial polynomial equation in \(\phi\) and its derivatives with coefficients from \(\mathbb{C}[z].\) The authors also give a geometric sufficient criterion for \(\text{itlog}(f)\) to be differentially transcendental over the ring of convergent power series. The results of the paper apply, in particular, to the exponential generating function of a sequence arising from work of Shadrin and Zvonkine on Hurwitz numbers.