Computational Methods and Function Theory 2 (2002), No. 1, 67--85
Copyright Heldermann Verlag 2002
Katsuya Ishizaki
ishi@nit.ac.jp, Department of Mathematics, Nippon Institute of Technology, 4-1 Gakuendai Miyashiro, Minamisaitama, Saitama 345-8501, Japan.
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 $n\geq9$ and the non-existence of transcendental entire solutions if $n\geq7$. 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 $h$.
Keywords: Meromorphic functions, Fermat type functional equations, value distribution theory, complex differential equations.
MSC 2000: 30D35.
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