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

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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.

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  1. Department of Mathematics, Nippon Institute of Technology, 4-1 Gakuendai Miya-shiro, Minamisaitama, Saitama, 345-8501, Japan

    Katsuya Ishizaki

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  1. Katsuya Ishizaki
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Correspondence to Katsuya Ishizaki.

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Dedicated to Professor Ilpo Laine on the occasion of his 60th birthday

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Ishizaki, K. A Note on the Functional Equation f n + g n + h n = 1 and Some Complex Differential Equations. Comput. Methods Funct. Theory 2, 67–85 (2003). https://doi.org/10.1007/BF03321010

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