Existence of meromorphic solutions of Fermat-type functional equations

Abstract:

      The functional equation $f(z)^n+g(z)^n=1$ can be interpreted as the Fermat-type equations over function field. In this paper, by using Nevanlinna theory of meromorphic functions, we investigate the existence of meromorphic solutions of hyper-order strictly less than 1 to the Fermat-type functional equation $\left(a_0f\left(z\right)+a_1f\left(z+c\right)\right)^3+\left(b_0f\left(z\right)+b_1f\left(z+c\right)\right)^3=e^{\alpha z+\beta},$ where $a_0$, $a_1$, $b_0$, $b_1$, $\alpha$, $\beta$, $c$ are complex constants and $c\neq0$.