Solutions of certain Fermat-type partial differential-difference equations

TL;DR

This paper characterizes entire and meromorphic solutions of a Fermat-type partial differential-difference equation in multiple complex variables, extending previous results from two to m variables and solving an open problem in the field.

Contribution

It generalizes existing results from two variables to m variables and provides a positive answer to an open problem posed by Xu and Wang.

Findings

Classifies solutions of the Fermat-type PDE-Difference equations in ^m.

Extends previous results from ^2 to ^m.

Provides numerous examples illustrating the solutions.

Abstract

The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial z_j}\right)^{m_1} + f^{m_2}(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m ) = 1,\] where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1, c_2, \ldots, c_m)\in \mathbb{C}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from $\mathbb{C}^2$ to $\mathbb{C}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings.