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

Let x be an indeterminate over C. We investigate solutions b(s, x) = Sigma(n >= 0)beta(n)(s)x(n), beta(n) : C -> C, n >= 0, of the second cocycle equation beta(s + t, x) = beta(s, x) alpha(t, F(s, x)) + beta(t, F(s, x)), s, t is an element of C, (Co2) in C[x], the ring of formal power series over C, where (F(s, x))(s is an element of C) is an iteration group of type II, i.e. it is a solution of the translation equation F(s + t, x) = F(s, F(t, x)), s, t is an element of C, (T) of the form F(s, x) equivalent to x + c(k)(s)x(k) mod x(k+1), where k >= 2 and c(k) not equal 0 is necessarily an additive function. Moreover, (alpha(s, x))(s is an element of C) is a solution of the first cocycle equation alpha(s + t, x) = alpha(s, x)alpha(t, F(s, x)), s, t is an element of C. (Co1) Using the method of 'formal functional equations' applied already for the study of F and alpha in previous manuscripts we obtain a formal version of the second cocycle equation in the ring (C[S, U, sigma])[[x]]. We solve this equation in a completely algebraic way, by deriving formal differential equations. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character of these polynomials depending on certain parameters, the coefficients of the generator K of a formal first cocycle and the coefficients of three generators N-1, N-2, N-3 of a formal second cocycle for iteration groups of type II.