Entire solutions of system of Fermat-type difference and partial   differential-difference equations in $ \mathbb{C}^2 $

Goutam Haldar

arXiv:2201.10560·math.CV·January 27, 2022

Entire solutions of system of Fermat-type difference and partial differential-difference equations in $ \mathbb{C}^2 $

Goutam Haldar

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TL;DR

This paper investigates entire solutions of Fermat-type difference and differential-difference equations in complex variables, extending previous results and providing explicit examples of solutions.

Contribution

It offers new existence results and explicit forms of solutions for complex Fermat-type systems, improving upon prior theorems in the field.

Findings

01

Established conditions for finite order entire solutions.

02

Extended previous theorems with new solution forms.

03

Provided explicit examples illustrating the solutions.

Abstract

In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations ${f_{1} (z)^{2} + (Δ_{c} f_{2} (z))^{2} = 1 f_{2} (z)^{2} + (Δ_{c} f_{1} (z))^{2} = 1,$ ${a_{1}^{2} f_{1} (z)^{2} + (a_{2} f_{2} (z + c) + a_{3} f_{2} (z))^{2} = 1 a_{1}^{2} f_{2} (z)^{2} + (a_{2} f_{1} (z + c) + a_{3} f_{1} (z))^{2} = 1,$ ${(a_{1} f_{1} (z + c) + a_{2} f_{1} (z))^{2} + (a_{3} f_{2} (z + c) + a_{4} f_{2} (z))^{2} = 1 (a_{1} f_{2} (z + c) + a_{2} f_{2} (z))^{2} + (a_{3} f_{1} (z + c) + a_{4} f_{1} (z))^{2} = 1,$ and ${(\partial^{I} f_{1} (z) + \partial^{J} f_{1} (z))^{n_{1}} + f_{2} (z + c)^{m_{1}} = 1 (\partial^{I} f_{2} (z) + \partial^{J} f_{2} (z))^{n_{2}} + f_{1} (z + c)^{m_{2}} = 1$ in several complex variables. Some of our results are improvements and extensions of the previous theorems given by Zheng-Xu \cite{Zheng-Xu & Analysis math & 2021}, Xu-Cao \cite{Xu & Cao &…

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Taxonomy

TopicsMeromorphic and Entire Functions · Advanced Differential Equations and Dynamical Systems · Nonlinear Differential Equations Analysis