Properties of Schwarzian difference equations

Abstract

We consider the Schwarzian type difference equation $$ \Big[\frac{\Delta^{3}f(z)}{\Delta f(z)}-\frac{3}{2} \Big(\frac{\Delta^{2}f(z)}{\Delta f(z)}\Big)^{2}\Big]^{k}=R(z), $$ where R(z) is a nonconstant rational function. We study the existence of rational solutions and value distribution of transcendental meromorphic solutions with finite order of the above equation.

Keywords

• Schwarzian difference equation
• rational solution
• value distribution