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On the Fermat-type equation \({f^3(z)+f^3(z+c)=1}\). (English) Zbl 1378.30015

I. N. Baker [Proc. Am. Math. Soc. 17, 819–822 (1966; Zbl 0161.35203)] has shown that each pair of solutions \(F\) and \(G\) to the equation \[ F^3 + G^3 = 1, \] such that \(F\) and \(G\) are meromorphic in the complex plane, must satisfy \(F(z)=f(h(z))\) and \(G(z)=\eta g(h(z))\), where \(f\) and \(g\) can be expressed in terms of the Weierstrass elliptic function, \(h\) is an entire function and \(\eta\) is a cube root of unity. In the paper under review the authors apply the result of Baker to show that the difference equation \[ F(z)^3 + F(z+c)^3 = 1, \] where \(c\in\mathbb{C}\setminus\{0\}\), does not have any non-constant meromorphic solutions of finite order. As an application, the authors consider a problem on unique range sets of finite-order meromorphic functions and their differences.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A10 Additive difference equations
39B32 Functional equations for complex functions

Citations:

Zbl 0161.35203
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Full Text: DOI

References:

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