Estimating the \(k\)th coefficient of \((f(z))^{n}\) when \(k\) is not too large (Q996915)

Assume that (i) the radius of covergence of the series \(f(z)=1+ \sum^\infty_{k=1} a_k(f)z^k\) is positive, (ii) all the coefficients \(a_k(f)\) are real, (ii) \(f\) is strongly positive, that is, for all sufficiently small \(r>0\), \(|f(re^{i\theta})| <f(r)\), \(0<\theta<2\pi\). Let \([f(z)]^n=1+ \sum^\infty_{k=1}a_k^{(n)}(f)z^k\). This article is a continuation of previous papers [Ergodic Theory Dyn. Syst. 14, No. 1, 23--51 (1994; Zbl 0840.26011), Proc. Am. Math. Soc. 123, No. 10, 2999--3007 (1995; Zbl 0933.26004)] by the same author. Theorem 1 extends results concerning the asymptotic expansion of \(a_k^{(n)}(f)\) while Theorem 2 provides asymptotic estimations for \(a_k^{(n)}(f)\), provided \(k\) belongs to a specified range. Some examples are given.