Under certain conditions on a function \(f\) it is shown that the \((kn)\)-th iterates of \(f\) have the asymptotic structure \(f^{(kn)}(x/n)=\sum_{i=1}^ r (nk)^{-i} f_ i(kx)+o(n^{-r})\) for \(n\to\infty\), where \(k\), \(n\) are natural numbers and \(| x|\) is sufficiently small. Using the complete complex iterate of \(f\), the result is also transferred to complex \(k\). The case \(r=3\) was already treated by the reviewer [ibid. 20, No. 1/2, 424-430 (1991; Zbl 0736.39001)].