Address.
Thomas W. Mueller,
School of Mathematical Sciences,
Queen Mary and Westfield College,
Mile End Road, London E1 4NS, England
Jan-Christoph
Schlage-Puchta,
Mathematisches Institut, Universitaet Freiburg,
Eckerstrasse 1, 79104 Freiburg, Germany
E-mail. T.W.Muller@qmul.ac.uk
E-mail. jcp@mathematik.uni-freiburg.de
Abstract.
We introduce the concept of asymptotic stability for a set of complex
functions analytic
around the origin, implicitly contained in an earlier paper of the
first mentioned author (``Finite group actions and asymptotic expansion
of $e^{P(z)}$",
Combinatorica 17 (1997), 523 -- 554). As a consequence of our main
result we find that the collection of entire functions
$\exp(\mathfrak{P})$ with $\mathfrak{P}$ the set of all real
polynomials $P(z)$ satisfying Hayman's condition
$[z^n]\exp(P(z))>0\,(n\geq n_0)$ is asymptotically
stable. This answers a question raised in loc. cit.
AMSclassification. 30B10.
Keywords. Power series, coefficients, asymptotic expansion.