On meromorphic solutions of functional equations of Fermat type

Abstract

Take complex numbers $a_j,b_j$, $(j=0,1,2)$ such that $c\neq0$ and {\rm rank} ( {ccc} a_{0} & a_{1} & a_{2} b_{0} & b_{1} & b_{2} )=2. We show that if the following functional equation of Fermat type \left\{a_{0}f(z)+a_{1}f(z+c)+a_{2}f'(z)\right\}^3+\left\{b_{0}f(z)+b_{1}f(z+c)+b_{2}f'(z)\right\}^3=e^{\alpha z+\beta} has meromorphic solutions of finite order, then it has only entire solutions of the form $f(z)=Ae^{\frac{\alpha z+\beta}{3}}+Ce^{Dz},$ which generalizes the results in {19} and {14}.