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Wiman-Valiron method for difference equations

  • K. Ishizaki (a1) and N. Yanagihara (a2)
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 χ < 1/2, then we have log M (r, f) = Lrχ(1 + o(1)) with a constant L > 0.

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References
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[1] Bank, S. B. and Kaufman, R. P., An extension of Hölder’s theorem concerning the Gamma function, Funkcialaj Ekvacioj, 19 (1976), 5363.
[2] Bergweiler, W., Ishizaki, K., and Yanagihara, N., Growth of meromorphic solutions of some functional equations I, Aequationes Math., 63 (2002), 140151.
[3] Boas, R. P. Jr., Entire functions, Academic Press Inc., New York, 1954.
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[5] Hayman, W. K., The local growth of power series: A survey of the Wiman–Valiron method, Canad. Math. Bull., 17 (1974), 317358.
[6] Helmrath, W. and Nikolaus, J., Ein elementarer Beweis bei der Anwendung der Zentralindexmethode auf Differentialgleichungen, Complex Variables Theory Appl., 3 (1984), 253262.
[7] Kövari, T., On the Borel exceptional values of lacunary integral functions, J. Analyse Math., 9 (1961), 71109.
[8] Laine, I., Nevanlinna theory and complex differential equations, W. Gruyter, Berlin–New York, 1992.
[9] Nörlund, N.E., Vorlesungen über Differenzenrechnung, Chelsea Publ., New York, 1954.
[10] Wittich, H., Neuere Untersuchungen über eindeutige analytische Funktionen, Springer-Verlag, 1955.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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