Computational Methods and Function Theory 5 (2005), No. 1, 77--88
Copyright Heldermann Verlag 2005
Ilpo Laine
ilpo.laine@joensuu.fi , University of Joensuu, Department of Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland.
Jarkko Rieppo
Jarkko.Rieppo@jns.fi , University of Joensuu, Department of Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland.
Heli Silvennoinen
heli.silvennoinen@jnor.joensuu.fi , University of Joensuu, Department of Mathematics, P.O. Box 111, FI-80101 Joensuu, Finland.
Halburd and Korhonen have shown that the existence of sufficiently many meromorphic solutions of finite order is enough to single out a discrete form of the second Painlevé equation from a more general class f(z+1)+f(z-1)=R(z,f) of complex difference equations. A key lemma in their reasoning is to show that f(z) has to be of infinite order, provided that degfR(z,f)≤2 and that a certain growth condition for the counting function of distinct poles of f(z) holds. In this paper, we prove a generalization of this lemma to higher order difference equations of more general type. We also consider related complex functional equations.
Keywords: Complex difference equation, complex functional equation, Nevanlinna theory, value distribution theory.
MSC 2000: Primary 39A10; Secondary 30D35, 39A12.
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