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

Abstract

We derive an asymptotic expansion asn→∞for a large range of coefficients of(f(z))n, wheref(z)is a power series satisfying|f(z)|<f(|z|)forz∈ℂ,z∉ℝ+. Whenfis a polynomial and the two smallest and the two largest exponents appearing infare 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 offhas only positive coefficients.