Painlevé III and V Types Differential Difference Equations

In this paper, we show that if the equations w(z+1)w(z-1)+a(z)w′(z)w(z)=P(z,w(z))Q(z,w(z)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w(z+1)w(z-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$\end{document}and (w(z)w(z+1)-1)(w(z)w(z-1)-1)+a(z)w′(z)w(z)=P(z,w(z))Q(z,w(z)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (w(z)w(z+1)-1)(w(z)w(z-1)-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$\end{document}where a(z) is rational, P(z, w) and Q(z, w) are coprime polynomials of w(z) with rational functions coefficients, have a non-rational meromorphic solution with hyper-order less than one, then the degrees of the numerator and denominator on the right sides of the equations have to meet certain conditions.