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Permutable Entire Functions Satisfying Algebraic Differential Equations

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Abstract

We show that if f and g are transcendental entire functions such that f(g) = g(f), then f satisfies an algebraic differential equation if and only if g does.

Keywords

Permutable commuting iteration factorization differential equation 

2000 MSC

30D05 34M05 39B12 

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References

  1. 1.
    I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1959), 121–163.CrossRefGoogle Scholar
  2. 2.
    D. Bargmann, Iteration holomorpher Funktionen, Dissertation, Christian-Albrechts-Universität zu Kiel, 1996.Google Scholar
  3. 3.
    W. D. Brownawell, On the factorization of partial differential equations, Canad. J. Math. 39 (1987), 825–834.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. Clunie, The composition of entire and meromorphic functions, in: H. Shankar (ed.), Mathematical essays dedicated to A. J. Macintyre, Athens, Ohio, Ohio University Press 1970, 75–92.Google Scholar
  5. 5.
    A. E. Eremenko, Some functional equations connected with the iteration of rational functions (in Russian), Algebra i Analiz 1 (1989), 102–116; translation in: Leningrad Math. J. 1 (1990), 905–919.MathSciNetGoogle Scholar
  6. 6.
    P. Fatou, Sur l’itération analytique et les substitutions permutables, J. Math. (9) 2 (1923), 343–384.Google Scholar
  7. 7.
    F. Gross and C. F. Osgood, A simpler proof of a theorem of Steinmetz, J. Math. Anal. Appl. 143 (1989), 490–496.MathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Gross and C. F. Osgood, An extension of a theorem of Steinmetz, J. Math. Anal. Appl. 156 (1991), 290–294.MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Gross and C. F. Osgood, Finding all solutions related to Steinmetz’s theorem, J. Math. Anal. Appl. 164 (1992), 417–421.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    W. K. Hayman, Meromorphic Functions, Oxford, Clarendon Press, 1964.MATHGoogle Scholar
  11. 11.
    T. Kobayashi, Permutability and unique factorizability of certain entire functions, Kodai Math. J. 3 (1980), 8–25.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    G. Julia, Mémoire sur la permutabilité des fractions rationnelles, Ann. Sci. Ecole Norm. Sup. (3) 39 (1922), 131–215.MathSciNetMATHGoogle Scholar
  13. 13.
    T. W. Ng, Permutable entire functions and their Julia sets, Math. Proc. Cambridge Philos. Soc. 131 (2001), 129–138.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    A. Ostrowski, Uber Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), 241–298.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    G. Pólya, On an integral function of an integral function, J. London Math. Soc. 1 (1926), 12–15.MATHCrossRefGoogle Scholar
  16. 16.
    J. F. Ritt, Transcendental transcendency of certain functions of Poincaré, Math. Ann. 95 (1925/26), 671–682.MathSciNetCrossRefGoogle Scholar
  17. 17.
    N. Steinmetz, Über faktorisierbare Lösungen gewöhnlicher Differentialgleichungen, Math. Z. 170 (1980), 169–180.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    C.-C. Yang and H. Urabe, On permutability of certain entire functions, J. London Math. Soc. (2) 14 (1976), 153–159.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    W. Xiong, On the permutability of entire functions, Nanjing Daxue Xuebao Shuxue Bannian Kan 17 (2000), 56–63.MathSciNetMATHGoogle Scholar
  20. 20.
    J. H. Zheng, On permutability of periodic entire functions, J. Math. Anal. Appl. 140 (1989), 262–269.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    J. H. Zheng and C.-C. Yang, Permutability of entire functions, Kodai Math. J. 15 (1992), 230–235.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    J. H. Zheng and C.-C. Yang, On the permutability of entire functions, J. Math. Anal. Appl. 167 (1992), 152–159.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    J. H. Zheng and Z. Z. Zhou, Permutability of entire functions satisfying certain differential equations, Tohoku Math. J. (2) 40 (1988), 323–330.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2008

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

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