On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

• R. Mandal Raiganj University
• R. Biswas Raiganj University
Keywords: System, Fermat-type equation, Entire solution, Several complex variables, Partial differential-difference equation, Nevanlinna theory

Abstract

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Author Biographies

R. Mandal, Raiganj University

Assistant Professor,

Department of Mathematics, Raiganj University, Raiganj, West Bengal, India

R. Biswas, Raiganj University

Department of Mathematics, Raiganj University
Raiganj, West Bengal, India

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Published
2024-06-19
How to Cite
Mandal, R., & Biswas, R. (2024). On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations. Matematychni Studii, 61(2), 195-213. https://doi.org/10.30970/ms.61.2.195-213
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