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On meromorphic solutions of non-linear difference equations. (English) Zbl 1401.30032

Summary: In this paper, using the theory of linear algebra, we investigate the non-linear difference equation of the following form in the complex plane: \[ f(z)^n + p(z)f(z+\eta ) = \beta _1e^{\alpha _1z}+\beta _2e^{\alpha _2z}+\cdots +\beta _se^{\alpha _sz}, \] where \(n\), \(s\) are the positive integers, \(p(z)\not \equiv 0\) is a polynomial and \(\eta , \beta _1,\dots , \beta _s, \alpha _1, \dots , \alpha _s\) are the constants with \(\beta _1 \dots \beta _s\alpha _1 \dots \alpha _s\neq 0\), and show that this equation just has meromorphic solutions with hyper-order at least one when \(n\geq 2+s\). Other cases are also obtained.

MSC:

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