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Finite order meromorphic solutions of linear difference equations. (English) Zbl 1226.30032

Summary: We mainly investigate the growth and the value distribution of meromorphic solutions of the linear difference equation \[ a_{n}(z)f(z+n)+\cdots +a_{1}(z)f(z+1)+a_{0}(z)f(z)=b(z), \] where \(a_{0}(z),a_{1}(z),\dots,a_{n}(z),b(z)\) are entire functions such that \(a_{0}(z)a_{n}(z)\not\equiv 0\). For a finite order meromorphic solution \(f(z)\), some interesting results on the relation between \(\rho=\rho(f)\) and \(\lambda_{f}=\max\{\lambda(f),\lambda(1/f)\}\), are proved. Examples are provided for our results.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A13 Difference equations, scaling (\(q\)-differences)
39A22 Growth, boundedness, comparison of solutions to difference equations
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