Computational Methods and Function Theory 1 (2001), No. 1, 27--39
Copyright Heldermann Verlag 2001
Janne Heittokangas
heittoka@cc.joensuu.fi, University of Joensuu, Department of Mathematics, P. O. Box 111, FIN-80101 Joensuu, Finland.
Risto Korhonen
rkorhone@cc.joensuu.fi, University of Joensuu, Department of Mathematics, P. O. Box 111, FIN-80101 Joensuu, Finland.
Ilpo Laine
Ilpo.Laine@joensuu.fi, University of Joensuu, Department of Mathematics, P. O. Box 111, FIN-80101 Joensuu, Finland.
Jarkko Rieppo
jrieppo@pelu.jns.fi, University of Joensuu, Department of Mathematics, P. O. Box 111, FIN-80101 Joensuu, Finland.
Kazuya Tohge
tohge@t.kanazawa-u.ac.jp, Kanazawa University, Faculty of Technology, 2-40-20 Kodatsuno, Kanazawa 920-8667, Japan.
In a recent paper \cite{AHH}, Ablowitz, Halburd and Herbst applied Nevanlinna theory to prove some results on complex difference equations remi\-nis\-cent 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 $\deg_y R(z,y)\leq 2$. Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result (see Theorem~\ref{jarkkothm2}) is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.
Keywords: Complex difference equation, value distribution, Nevanlinna characteristic, Borel exceptional values.
MSC 2000: Primary 39A10; Secondary 30D35, 39A12.
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