Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Abstract

Fermat type functional equation with four terms

$$\begin{aligned} f(z)^{n}+g(z)^{n}+h(z)^{n}+k(z)^{n}=1 \end{aligned}$$

is difficult to solve completely even if \(n=2,3\), in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation

$$\begin{aligned} f(z_{1},z_{2})^{2}+\left( \frac{\partial f(z_{1},z_{2})}{\partial z_{1}}\right) ^{2}+g(z_{1},z_{2})^{2}+\left( \frac{\partial g(z_{1},z_{2})}{\partial z_{1}}\right) ^{2}=1 \end{aligned}$$

in \(\mathbb {C}^{2}\). In addition, we consider the Bi-Fermat type cubic difference equation

$$\begin{aligned} f(z)^{3}+g(z)^{3}+f(z+c)^{3}+g(z+c)^{3}=1 \end{aligned}$$

in \(\mathbb {C}\) and obtain partial meromorphic solutions on the above equation.