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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

$y\left(z+1\right)+y\left(z-1\right)=R\left(z,y\right)$

with $R\left(z,y\right)$ rational in both arguments admits a transcendental meromorphic solution of finite order, then ${deg}_{y}R\left(z,y\right)\le 2$.

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.

##### MSC:
 39A10 Additive difference equations 34M05 Entire and meromorphic solutions (ODE) 39A12 Discrete version of topics in analysis 30D35 Distribution of values (one complex variable); Nevanlinna theory