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Remarks on Complex Difference Equations

Remarks on Complex Difference Equations 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2005
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321087
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 1, 2005

Keywords: Complex difference equation; complex functional equation; Nevanlinna theory; value distribution theory; 39A10; 30D35; 39A12

References