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
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.
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.