On Existence of Solutions of Difference Riccati Equation

Consider the difference Riccati equation xxxx, where A, B, C, D are meromorphic functions, we give its solution family with one-parameter H(f(z))={f0(z),f(z)=(f1(z)−f0(z))(f2(z)−f0(z))Q(z)(f2(z)−f1(z))+(f2(z)−f0(z))+f0(z)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\left( {f\left( z \right)} \right) = \left\{ {{f_0}\left( z \right),f\left( z \right) = \frac{{\left( {{f_1}\left( z \right) - {f_0}\left( z \right)} \right)\left( {{f_2}\left( z \right) - {f_0}\left( z \right)} \right)}}{{Q\left( z \right)\left( {{f_2}\left( z \right) - {f_1}\left( z \right)} \right) + \left( {{f_2}\left( z \right) - {f_0}\left( z \right)} \right)}} + {f_0}\left( z \right)} \right\}$$\end{document} , where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f0(z), f1(z), f2(z) are its three distinct meromorphic solutions.