On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

In this paper, we investigate the existence and specific form of finite order transcendental entire solutions of certain equations including a Fermat-type functional first-order linear difference equation in \(\mathbb {C}^n\), \(n\geqslant 2\) and a kth order partial differential difference equation in \(\mathbb {C}^2\). The paper builds upon the previous works of Xu and Cao (Mediterr J Math 15:1–14, 2018; Mediterr J Math 17:1–4, 2020) and Haldar (Mediterr J Math 20: 50, 2023) whose results are extended and further developed in this study. We exhibit several examples to demonstrate the precision and applicability of our results to illustrate how our findings can be utilized in different scenarios or problem contexts. Towards the end of the paper, in the last section, we discuss some relevant questions that have emerged from one of the examples in the paper which also suggest potential directions for further research.