Geodesic
Asymptotic stability for sets of polynomials
Archivum mathematicum, Tome 41 (2005) no. 2, pp. 151-155 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 30B10, 30D15
Keywords: power series; coefficients; asymptotic expansion 
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Müller, Thomas W.; Schlage-Puchta, Jan-Christoph. Asymptotic stability for sets of polynomials. Archivum mathematicum, Tome 41 (2005) no. 2, pp. 151-155. http://geodesic.mathdoc.fr/item/ARM_2005_41_2_a2/

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[3] Müller T.: Finite group actions and asymptotic expansion of $e^{P(z)}$. Combinatorica 17 (1997), 523–554. | MR

[4] Stanley R. P.: Enumerative Combinatorics. vol. 2, Cambridge University Press, 1999. | MR | Zbl