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A Note on the Functional Equation f n + g n + h n = 1 and Some Complex Differential Equations

  • Katsuya Ishizaki
Article

Abstract

We consider entire and meromorphic solutions of the functional equation f n + g n + h n = 1. We give new proofs for the known results about the non-existence of transcendental meromorphic solutions for n ≥ 9 and the non-existence of transcendental entire solutions if n ≥ 7. It is shown that if there exist transcendental meromorphic functions f, g and h satisfying the functional equation f 8 + g 8 + h 8 = 1, then f, g and h satisfy the differential equation W(f 8, g 8, h 8) = a(z)(f(z)g(z)h(z))6, where a(z) is a small function with respect to f, g and h.

En]Keywords

Meromorphic functions Fermat type functional equations value distribution theory complex differential equations 

2000 MSC

30D35 

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Copyright information

© Heldermann  Verlag 2002

Authors and Affiliations

  1. 1.Department of MathematicsNippon Institute of TechnologyMinamisaitama, SaitamaJapan

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