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Remarks on complex difference equations. (English) Zbl 1093.39018
Authors’ abstract: R. Halburd and R. Korhonen [Existence of finite order meromorphic solutions as a detector of integrability in difference equations. (to appear)] 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 deg f 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.
MSC:
39A12Discrete version of topics in analysis
30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D35Distribution of values (one complex variable); Nevanlinna theory
39B12Iterative and composite functional equations
39B32Functional equations for complex functions
34M55Painlevé and other special equations; classification, hierarchies
39A10Additive difference equations
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