Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#

Abstract

In this article, we investigate the existence and the precise form of finite-order transcendental entire solutions of some system of Fermat-type quadratic binomial and trinomial shift equations in Cn{{\mathbb{C}}}^{n}. Our results are the generalizations of the results of [H. Y. Xu, S. Y. Liu, and Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483 (2020), 123641, 1–22, DOI: https://doi.org/10.1016/j.jmaa.2019.123641.] and [H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9.] to a large extent. Most interestingly, as a consequence of our main result, we have shown that the system of quadratic trinomial shift equation has no solution when it reduces to a system of quadratic trinomial difference equation. In addition, some examples relevant to the content of the article have been exhibited.

Keywords

• functions of several complex variables
• fermat-type shift equations
• entire solutions
• nevanlinna theory
• 30d35
• 39a45