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Keywords:
power series; coefficients; asymptotic expansion
power series; coefficients; asymptotic expansion
Summary:
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 $[z^n]\exp (P(z))>0\,(n\ge n_0)$ is asymptotically stable. This answers a question raised in loc. cit.
References:
[1] Dress A., Müller T.: Decomposable functors and the exponential principle. Adv. in Math. 129 (1997), 188–221. MR 1462733 | Zbl 0947.05002
[2] Hayman W.: A generalisation of Stirling’s formula. J. Reine u. Angew. Math. 196 (1956), 67–95. MR 0080749
[3] Müller T.: Finite group actions and asymptotic expansion of $e^{P(z)}$. Combinatorica 17 (1997), 523–554. MR 1645690
[4] Stanley R. P.: Enumerative Combinatorics. vol. 2, Cambridge University Press, 1999. MR 1676282 | Zbl 0945.05006
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