On the functional equation $f^n(z)+g^n(z)=e^{\alpha z+\beta}$

TL;DR

This paper investigates meromorphic solutions to specific functional equations involving powers and derivatives or shifts of functions over the complex plane, expanding understanding of their structure and solutions.

Contribution

It characterizes meromorphic solutions to the equations involving powers and derivatives or shifts, providing new insights into their form and existence.

Findings

Explicit descriptions of solutions for the equations

Conditions under which solutions exist

Extension of known results to more general equations

Abstract

We describe meromorphic solutions to the equations $f^n(z)+\left(f'\right)^n(z)=e^{\alpha z+\beta}$ and $f^n(z)+f^n(z+c)=e^{\alpha z+\beta}$ ($c\neq0$) over the complex plane $\mathbf{C}$ for integers $n\geq1$.