Entire and Meromorphic Solutions of the Functional Equation $$f^n+g^n+h^n=1$$
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Entire and Meromorphic Solutions of the Functional Equation f n + gn + hn = 1 and Differential Equations Katsuya Ishizaki1 · Naofumi Kimura1 Received: 15 June 2018 / Revised: 12 October 2018 / Accepted: 19 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract 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. Keywords Fermat-type functional equations · Polynomials · Rational functions · Meromorphic functions · Entire functions · Nevanlinna theory Mathematics Subject Classification 30D35 · 30D05
1 Introduction We consider the question of existence of non-constant entire and meromorphic functions f , g and h satisfying the Fermat-type functional equation f n + g n + h n = 1,
(1.1)
where n ≥ 2 is an integer. It is known for which positive integers n, the functional equation f n + g n = 1 has non-constant entire and meromorphic solutions f and g, see e.g. [1,2,5,7,11]. Hence, we assume in (1.1) that f n , g n and h n are linearly independent, otherwise (1.1) reduces to f n + g n = 1.
Communicated by Ilpo Laine.
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Katsuya Ishizaki [email protected] Naofumi Kimura [email protected]
1
Faculty of Liberal Arts, The Open University of Japan, Wakaba 2-11, Mihama-ku, Chiba 261-8586, Japan
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K. Ishizaki, N. Kimura
We know that there exist solutions of non-constant rational functions of (1.1) when n ≤ 5, and also know that there do not exist non-constant rational functions satisfying (1.1) when n ≥ 8. For the cases n ≤ 3, we know that there exist non-constant polynomial solutions of (1.1), and that there do not exist non-constant polynomial solutions of (1.1) when n ≥ 6. To our best knowledge, the cases n = 7 and 6 for solutions of non-constant rational functions, and the cases n = 5 and 4 for non-constant polynomial solutions are still open, see e.g. [6,7,13]. Example 1 We recall examples for n = 4 and n = 5. Let a(z) ≡ 0 be a meromorphic function, where ‘meromorphic’ means meromorphic in the whole complex plane C. The following functions 1 1 1 1 3 3 a(z) + , g(z) = √ , f (z) = √ a(z) − 4 4 a(z) a(z) −8 8 √ 4 h(z) = −1 a(z)2
(1.2)
satisfy (1.1) for n = 4, and functions √ √ 1 2+ 6 f (z) = (2 − 6)a(z) + 1 + , 3 a(z) √ √ 1 √ ( 6 − 2) + (3 2 − 2 3)i a(z) + 2 g(z) = 6 √ √ √ ( 6 + 2) − (3 2 + 2 3)i , − a(z) √ √ √ 1 ( 6 − 2) − (3 2 − 2 3)i a(z) + 2 h(z) = 6 √ √ √ ( 6 + 2) + (3 2 + 2 3)i − a(z)
(1.3)
(1.4)
(1.5)
satisfy (1.1) for n = 5, see [4,5,7,8]. Setting a rational function as a in Example 1, we ascertain that there exist solutions of non-constant rational functions of (1.1) when n = 4 and 5. Concerning transcendental cases for (1.1), there exist transcendental meromorphic solutions of (1.1) when n ≤ 6, see [3]; we also know that there is no transcendental meromorphic solutions of (1.1) when n ≥ 9, see [7,10]. For the case n
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