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Zbl 1013.39001
Heittokangas, Janne; Korhonen, Risto; Laine, Ilpo; Rieppo, Jarkko; Tohge, Kazuya
Complex difference equations of Malmquist type. (English)
[J] Comput. Methods Funct. Theory 1, No.1, 27-39 (2001). ISSN 1617-9447

Summary: {\it M. J. Ablowitz}, {\it R. Halburd} and {\it 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 $\deg_yR(z,y)\le 2$.\par 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 2000:: *39A10 Difference equations
34M05 Entire and meromorphic solutions
39A12 Discrete version of topics in analysis
30D35 Distribution of values (one complex variable)

Keywords: complex difference equation; value distribution; Nevanlinna characteristic; Borel exceptional values; Malmquist theorem

Citations: Zbl 0956.39003

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