Analytic solutions of systems of functional equations. (Q1413694)

For a function \(f\) denote by \(f^n\) its \(n\)-th iterate. For a function \(H\) of \(n\) variables put \(\Psi _H (f)(z)=H(z,f(z), f^2(z),\dots,f^{n-1}(z)).\) The author studies a system of functional equations involving two functions \(f\) and \(g\) as well as the functions \(\Psi_{H_j}(f)\) and \(\Psi_{K_j}(g)\) with certain functions \(H_j\) and \(K_j\), \(j= 2,3,\dots,n.\) It is shown that this system has a unique solution under certain conditions.