Nagoya Mathematical Journal

Wiman-Valiron method for difference equations

K. Ishizaki and N. Yanagihara

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Abstract

Let $f(z)$ be an entire function of order less than $1/2.$ We consider an analogue of the Wiman-Valiron theory rewriting power series of $f(z)$ into binomial series. As an application, it is shown that if a transcendental entire solution $f(z)$ of a linear difference equation is of order $\chi < 1/2,$ then we have %$\chi$ is obtained from the Newton polygon for the equation, and $\log M(r,f) = Lr^{\chi}(1 + o(1))$ with a constant $L > 0.$

Article information

Source
Nagoya Math. J., Volume 175 (2004), 75-102.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114632096

Mathematical Reviews number (MathSciNet)
MR2085312

Zentralblatt MATH identifier
1070.39002

Subjects
Primary: 39A05: General theory
Secondary: 30D35: Distribution of values, Nevanlinna theory

Citation

Ishizaki, K.; Yanagihara, N. Wiman-Valiron method for difference equations. Nagoya Math. J. 175 (2004), 75--102. https://projecteuclid.org/euclid.nmj/1114632096


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References

  • S. B. Bank and R. P. Kaufman, An extension of Hölder's theorem concerning the Gamma function , Funkcialaj Ekvacioj, 19 (1976), 53–63.
  • W. Bergweiler, K. Ishizaki, and N. Yanagihara, Growth of meromorphic solutions of some functional equations I , Aequationes Math., 63 (2002), 140–151.
  • R. P. Boas, Jr., Entire functions, Academic Press Inc., New York (1954).
  • G. Gundersen, G. Steinbart, M. Enid and S. Wang, The possible orders of solutions of linear differential equations with polynomial coefficients , Trans. Amer. Math. Soc., 350 (1998), 1225–1247.
  • W. K. Hayman, The local growth of power series: A survey of the Wiman–Valiron method , Canad. Math. Bull., 17 (1974), 317–358.
  • W. Helmrath and J. Nikolaus, Ein elementarer Beweis bei der Anwendung der Zentralindexmethode auf Differentialgleichungen , Complex Variables Theory Appl., 3 (1984), 253–262.
  • Kövari, T., On the Borel exceptional values of lacunary integral functions , J. Analyse Math., 9 (1961), 71–109.
  • Laine, I., Nevanlinna theory and complex differential equations, W. Gruyter, Berlin–New York (1992).
  • Nörlund, N.E., Vorlesungen über Differenzenrechnung, Chelsea Publ., New York (1954).
  • Wittich, H., Neuere Untersuchungen über eindeutige analytische Funktionen, Springer-Verlag (1955).