Solutions to some generalized Fermat-type differential-difference equations

Abstract

The main purpose of this article is to study Fermat-type complex differential-difference equations $ f^{(k)}(z)^{2}+[\alpha f(z+c)-\beta f(z)]^{2} = R(z) $. Our results improve some results due to Wang–Xu–Tu [AIMS. Mathematics, 2020], Zhang [Bull. Korean. Math. Soc, 2018], and Long–Qin [Applied Mathematics-A Journal of Chinese Universities, 2024]. Moreover, we provide some examples to show the existence of the solutions.

Keywords

• meromorphic functions
• fermat-type equation
• differential polynomials
• nevanlinna theory
• differential-difference equation