Computational Methods and Function Theory

, Volume 19, Issue 1, pp 157–172 | Cite as

Entire and Meromorphic Solutions of the Functional Equation \(f^n+g^n+h^n=1\) and Differential Equations

  • Katsuya IshizakiEmail author
  • Naofumi Kimura


In this paper, we study Fermat-type functional equations \(f^n+g^n+h^n=1\) in the complex plane. Alternative proofs of the known results for entire and meromorphic solutions of such equations are given. Moreover, some conditions on degrees of polynomial solutions are given.


Fermat-type functional equations Polynomials Rational functions Meromorphic functions Entire functions Nevanlinna theory 

Mathematics Subject Classification

30D35 30D05 



The authors would like to thank Professor Gary G. Gundersen for his valuable suggestions. The authors would also like to thank the referees for their helpful comments. This research has financially been supported by the discretionary budget (2017) of the President of the Open University of Japan.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Liberal ArtsThe Open University of JapanChibaJapan

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