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Complex difference equations of Malmquist type. (English) Zbl 1013.39001
Summary: M. J. Ablowitz, R. Halburd and B. Herbst [Nonlinearity 13, No. 3, 889-905 (2000; Zbl 0956.39003)] applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation \[ y(z+ 1)+ y(z- 1)= R(z, y) \] with \(R(z, y)\) rational in both arguments admits a transcendental meromorphic solution of finite order, then \(\deg_yR(z,y)\leq 2\).
Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.

39A10 Additive difference equations
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
39A12 Discrete version of topics in analysis
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI Link
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