Asymptotic stability for sets of polynomials

Thomas W. Müller; Jan-Christoph Schlage-Puchta

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 2, page 151-155
  • ISSN: 0044-8753

Abstract

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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 ( 𝔓 ) with 𝔓 the set of all real polynomials P ( z ) satisfying Hayman’s condition [ z n ] exp ( P ( z ) ) > 0 ( n ≥ n 0 ) is asymptotically stable. This answers a question raised in loc. cit.

How to cite

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Müller, Thomas W., and Schlage-Puchta, Jan-Christoph. "Asymptotic stability for sets of polynomials." Archivum Mathematicum 041.2 (2005): 151-155. <http://eudml.org/doc/249491>.

@article{Müller2005,
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 $[z^n]\exp (P(z))>0\,(n\ge n_0)$ is asymptotically stable. This answers a question raised in loc. cit.},
author = {Müller, Thomas W., Schlage-Puchta, Jan-Christoph},
journal = {Archivum Mathematicum},
keywords = {power series; coefficients; asymptotic expansion; power series; coefficients; asymptotic expansion},
language = {eng},
number = {2},
pages = {151-155},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic stability for sets of polynomials},
url = {http://eudml.org/doc/249491},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Müller, Thomas W.
AU - Schlage-Puchta, Jan-Christoph
TI - Asymptotic stability for sets of polynomials
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 2
SP - 151
EP - 155
AB - 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.
LA - eng
KW - power series; coefficients; asymptotic expansion; power series; coefficients; asymptotic expansion
UR - http://eudml.org/doc/249491
ER -

References

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  1. Dress A., Müller T., Decomposable functors and the exponential principle, Adv. in Math. 129 (1997), 188–221. (1997) Zbl0947.05002MR1462733
  2. Hayman W., A generalisation of Stirling’s formula, J. Reine u. Angew. Math. 196 (1956), 67–95. (196) MR0080749
  3. Müller T., Finite group actions and asymptotic expansion of e P ( z ) , Combinatorica 17 (1997), 523–554. (1997) MR1645690
  4. Stanley R. P., Enumerative Combinatorics, vol. 2, Cambridge University Press, 1999. (1999) Zbl0945.05006MR1676282

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