On the formal first cocycle equation for iteration groups of type II

Let x be an indeterminate over ℂ. We investigate solutions αn : ℂ → ℂ, n ≥ 0, of the first cocycle equation in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation of the form F(s,x) ≡ x + ck(s)xk mod xk+1, where k ≥ 2 and ck ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions αn(s) of are polynomials in ck(s).