Transcendental transcendence of solutions of Schröder's equation associated with finite Blaschke products (Q1063723)

Let S be a finite Blaschke product with \(S(0)=0\) and \(S'(0)=s\), \(0<| s| <1\). Consider the three functional equations: \[ (S)\quad \phi (S(z))=s\phi (z),\quad | z| <1. \] \[ (E)\quad \Psi (S(z))=s^{\ell}(S'(z))^ m\Psi (z),\quad \ell,m\quad integers\quad not\quad both\quad 0 \] \[ (F)\quad \Psi (S(z))(S'(z))^ 2=\Psi (z)S'(z)+\lambda S''(z)\quad (\lambda \neq 0). \] The main result of the paper is that if \(\phi\) \(\not\equiv 0\) is meromorphic in the unit disk and satisfies (Schröder's) equation (S), then \(\phi\) satisfies no algebraic differential equation (ADE). An analysis modelled on \textit{A. Ostrowski}'s classical work on the gamma function [Math. Ann. 79, 286-288 (1919)] shows that if \(\phi\) did satisfy an ADE, then one of the equations (E) and (F) would have a (nontrivial) rational solution \(\Psi\). Essential use is made of the fact that S is a finite monic Blaschke product of degree at least 2 to show that no rational function \(\Psi\) satisfies (F), and that the only rational solution of (E) is the trivial one.