Value distribution of meromorphic solutions of certain difference Painlevé III equations. (English) Zbl 1446.39006
Summary: In this paper, we investigate the difference Painlevé III equations \(w(z+1)w(z-1)(w(z)-1)^{2}=w^{2}(z)-\lambda w(z)+\mu\) (\(\lambda\mu\neq 0\)) and \(w(z+1)w(z-1)(w(z)-1)^{2}=w^{2}(z)\), and obtain some results about the properties of the finite order transcendental meromorphic solutions. In particular, we get the precise estimations of exponents of convergence of poles of difference \(\Delta w(z)=w(z+1)-w(z)\) and divided difference \(\frac{\Delta w(z)}{w(z)}\), and of fixed points of \(w(z+\eta)\) (\(\eta\in C\setminus\{0\}\)).
MSC:
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 39A45 | Difference equations in the complex domain |
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
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