On meromorphic solutions of functional equations of Fermat type

TL;DR

This paper investigates the solutions of a specific Fermat-type functional equation involving meromorphic functions, showing that under certain conditions, solutions are necessarily entire and of a particular exponential form, extending previous results.

Contribution

It characterizes all finite-order meromorphic solutions of the Fermat-type functional equation, proving they are necessarily entire and of exponential form, generalizing earlier findings.

Findings

Solutions are entire functions of exponential form.

Finite-order meromorphic solutions only exist as entire functions.

The results extend previous classifications of solutions.

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}.