Notes on Fermat-Type Difference Equations

Abstract

We consider the existence problem of meromorphic solutions of the Fermat-type difference equation $$ \begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*} $$ where $p,q$ are positive integers, and h has few zeros and poles in the sense that $N(r,h) + N(r,1/h) = S(r,h)$ . As a particular case, we consider $h=e^g$ , where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense $T(r,h)=S(r,f)$ .