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

Abstract

We describe meromorphic solutions to the equations $f^n(z)+{\left(f^{\prime }\right)}^n(z)=e^{\alpha z+\beta }$ and $f^n(z)+f^n(z+c)=e^{\alpha z+\beta }$ ($c\ne 0$) over the complex plane $\mathbf{C}$ for integers $n\ge 1$.