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Remarks on Complex Difference Equations

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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 degf R(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.

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Correspondence to Ilpo Laine.

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Partially supported by the Academy of Finland grants 50981 and 210245, as well as by the Väisalä Fund of the Finnish Academy of Sciences and Letters.

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Laine, I., Rieppo, J. & Silvennoinen, H. Remarks on Complex Difference Equations. Comput. Methods Funct. Theory 5, 77–88 (2005).

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2000 MSC