On solutions of two categories of q-shift equations in two dimensional complex field

In this paper, we mainly focus on finding transcendental entire solutions of binomial and trinomial equations generated by q-shift operator in \({\mathbb {C}}^{2}\), to extend the results of Li and Xu (Axioms 126(10):1-19, 2021) and Zheng and Xu (Anal Math 48(1):199-226, 2022) completely in a different direction in terms of their q-shift counterpart. We have observed notable differences in the solutions derived from q-shift equations, including two different variants of q-difference equations, compared to the solutions obtained from the corresponding c-shift equations in \({\mathbb {C}}^{2}\). Our findings have been supported with several examples that illustrate these differences. Additionally, the introduction of two new lemmas not explored in existing literature further adds depth to the mathematical tools available for addressing such problems.