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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 $f\left(z+1\right)+f\left(z-1\right)=R\left(z,f\right)$ of complex difference equations. A key lemma in their reasoning is to show that $f\left(z\right)$ has to be of infinite order, provided that ${\text{deg}}_{f}R\left(z,f\right)\le 2$ and that a certain growth condition for the counting function of distinct poles of $f\left(z\right)$ 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:
 39A12 Discrete version of topics in analysis 30D05 Functional equations in the complex domain, iteration and composition of analytic functions 30D35 Distribution of values (one complex variable); Nevanlinna theory 39B12 Iterative and composite functional equations 39B32 Functional equations for complex functions 34M55 Painlevé and other special equations; classification, hierarchies 39A10 Additive difference equations
References:
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