International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 16, Pages 1003-1025
doi:10.1155/S0161171203205056

Asymptotic expansions and positivity of coefficients for large powers of analytic functions

Abstract

We derive an asymptotic expansion as n→∞ for a large range of coefficients of (f(z))n, where f(z) is a power series satisfying |f(z)|<f(|z|) for z∈ℂ, z∉ℝ+. When f is a polynomial and the two smallest and the two largest exponents appearing in f are consecutive integers, we use the expansion to generalize results of Odlyzko and Richmond (1985) on log concavity of polynomials, and we prove that a power of f has only positive coefficients.