Uniqueness of meromorphic solutions of the difference equation R1(z)f(z+1)+R2(z)f(z)=R3(z) $R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z)$

Abstract

Abstract This paper mainly concerns the uniqueness of meromorphic solutions of first order linear difference equations of the form * R1(z)f(z+1)+R2(z)f(z)=R3(z), $$ R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z), $$ where R1(z)≢0 $R_{1}(z)\not \equiv 0$, R2(z) $R_{2}(z)$, R3(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.

Keywords

• Meromorphic solutions
• Difference equations
• Uniqueness