Entire functions of several complex variables satisfying certain Fermat-type PDDEs

TL;DR

This paper investigates entire functions of multiple complex variables satisfying Fermat-type partial differential-difference equations, extending previous results from two variables to several variables and providing new solutions and examples.

Contribution

It generalizes earlier findings on quadratic binomial and trinomial PDEDEs from two to multiple complex variables, offering new solutions and broader applicability.

Findings

Solved Fermat-type PDEDEs for entire functions in several variables

Extended previous two-variable results to multiple variables

Provided examples illustrating the solutions

Abstract

In this paper, we solve certain Fermat-type partial differential-difference equations for finite order entire functions of several complex variables. These results are significant generalizations of some earlier findings, especially those of Haldar and Ahamed (Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $\mathbb{C}^2$, Anal. Math. Phys., 12 (2022)). In addition, the results improve the previous results from the situation with two complex variables to the situation with several complex variables. To support our results, we have included several examples.