Uniqueness of meromorphic solutions of the difference equation $R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z)$

This paper mainly concerns the uniqueness of meromorphic solutions of first order linear difference equations of the form * $$ R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z), $$ where $R_{1}(z)\not \equiv 0$ , $R_{2}(z)$ , $R_{3}(z)$ are rational functions. Our results indicate that the finite order transcendental meromorphic solution of equation (*) is mainly determined by its zeros and poles except for some special cases. Examples for the sharpness of our results are also given.