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On the Nevanlinna order of meromorphic solutions to linear analytic difference equations. (English) Zbl 1145.39300

Summary: For various classes of linear ordinary analytic difference equations with meromorphic coefficients, we study Nevanlinna order properties of suitable meromorphic solutions. For a large class of first-order equations with coefficient of order \(\rho \in [0, \infty )\), we explicitly construct meromorphic solutions of order \(\leq \rho + 1\). For higher-order equations with coefficients of order \(\rho \in [0, \infty )\), we show that meromorphic solutions with increase of order \(\leq \rho + 1\) in a certain strip have order \(\leq \rho + 1\). The assumptions made in the latter setting may seem quite restrictive, but they are satisfied for several classes of second-order difference equations that have been studied in recent years. The latter include Harper-type equations, ”reflectionless” equations, Askey-Wilson-type equations, and equations of relativistic Calogero-Moser type.

MSC:

39A10 Additive difference equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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