Zeros, Poles, and Fixed Points of Meromorphic Solutions of Difference Painlevé Equations

Abstract

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