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

This paper has involved the use of a variety of variations of the Fermat-type equation fⁿ (z) +gⁿ (z) =1, where n (2). 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 cases f^m₁+g^n₁=1;\\ f^m₂+g^n₂=1, cases\!\! if f, g are finite order entire functions satisfying the system, where m₁, m₂, n₁, n₂\1\. \ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: cases f^m₁+2 f g+g^n₁=1;\\ f^m₂+2 f g+g^n₂=1, cases \!\! where \0, 1\. In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on C², especially, those of Xu 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 Cⁿ. These results will extend the further study of this direction.