Solutions for certain Fermat-type PDDEs concerning an open problem of Xu and Wang

TL;DR

This paper investigates the existence and forms of finite order meromorphic and entire solutions to Fermat-type partial differential-difference equations in several complex variables, addressing an open problem posed by Xu and Wang.

Contribution

It extends the understanding of solutions to Fermat-type PDDEs in multiple variables, providing new existence results and illustrative examples for the open problem.

Findings

Existence of finite order meromorphic solutions established.

Explicit forms of solutions characterized.

Examples illustrating the theoretical results provided.

Abstract

The objective of this study is to ascertain the existence and forms of the finite order meromorphic and entire functions of several complex variables satisfying some certain Fermat-type partial differential-difference equations by considering the more general forms of the PDDEs in an open problem on $\mathbb{C}^2$ due to Xu and Wang (Notes on the existence of entire solutions for several partial differential-difference equations, Bull. Iran. Math. Soc., 47, 1477-1489 (2020)). We provide examples to illustrate the results.