International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 16, Pages 1003-1025

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

Valerio De Angelis

Department of Mathematics, Xavier University of Louisiana, 1 Drexel Drive, New Orleans 70125, LA, USA

Received 6 May 2002

Copyright © 2003 Valerio De Angelis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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, z+. 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.