Zeros, poles, and fixed points of meromorphic solutions of difference Painlevé equations. (English) Zbl 1474.39012

Lan, Shuang-Ting; Chen, Zong-Xuan

Abstr. Appl. Anal. 2014, Article ID 782024, 8 p. (2014).

Summary: In this paper, we mainly study the properties of transcendental meromorphic solutions \(f(z)\) of difference Painlevé equations \(w(z + 1) w(z - 1)(w(z) - 1) = \eta(z) w^2(z) - \lambda(z) w(z)\) and \(w(z + 1) w(z - 1)(w(z) -\)\(1) = \eta(z) w(z)\) and obtain precise estimations of the exponents of convergence of zeros, poles of \(\Delta f(z)\) and \(\Delta f(z) / f(z)\), and of fixed points of \(f(z + c)\) for any \(c \in \mathbb C\).

MSC:

39A12	Discrete version of topics in analysis
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies

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