Meromorphic solutions of difference equations originated from Schwarzan differential equation

Abstract

Let $f(z) $ be a meromorphic functions with finite order , $R(z)$ be a nonconstant rational function and $k$ is a positive integer. In this paper, we consider the difference equation originated from Schwarzian differential equation, which is of form
\begin{equation*}
\left[\Delta^{3}f(z)\Delta f(z)-\frac{3}{2}(\Delta^{2}f(z))^{2}\right]^{k}=R(z)(\Delta f(z))^{2k}.\eqno(*)
\end{equation*}
We investigate the uniqueness of meromorphic of equation (*) if $f$ shares three values with any meromrphic function. The exact forms of meromorphic solutions $f$ of equation (*) are also presented.