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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 y R(z,y)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:
39A10Additive difference equations
34M05Entire and meromorphic solutions (ODE)
39A12Discrete version of topics in analysis
30D35Distribution of values (one complex variable); Nevanlinna theory
References:
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