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Wiman-Valiron method for difference equations

Published online by Cambridge University Press:  22 January 2016

K. Ishizaki  and
N. Yanagihara
K. Ishizaki
Affiliation:
Department of Mathematics, Nippon Institute of Technology, 4-1 Gakuendai Miyashiro, Minamisaitama, Saitama-ken, 345-0826, Japan, ishi@nit.ac.jp
N. Yanagihara
Affiliation:
Minami-Iwasaki 671-18, Ichihara-City, Chiba-ken 290-0244, Japan, yanagihara-nm@icntv.ne.jp
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Abstract

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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 χ < 1/2, then we have log M (r, f) = Lrχ(1 + o(1)) with a constant L > 0.

Keywords

39B1030D35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 175 , 2004 , pp. 75 - 102
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

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