Existence of solutions of Fermat-type functional equations in ℂ

Fermat-type functional equations have long been regarded as challenging and intriguing problems in complex function theory. In this article, utilizing Nevanlinna theory, we first answer a conjecture posed by Zhang [Bull Korean Math Soc. 2018;55(1):51–61] in the study of Fermat-type functional equations. We then investigate the existence and explicit forms of solutions related to general quadratic functional equations of the form af2(z)+2αf(z)g(z)+bg2(z)+2βf(z)+2γg(z)+C=0. Moreover, we investigate the existence and solutions of Fermat-type difference equations with polynomial coefficients. The results obtained in this paper establish a unified and comprehensive framework for general quadratic functional equations, extending and encompassing a wide range of previous results on Fermat-type functional and difference equations.