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 binomialand trinomial equations generated by q-shift operator in C-2, to extend the results of Liand 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 haveobserved notable differences in the solutions derived fromq-shift equations, includingtwo different variants of q-difference equations, compared to the solutions obtainedfrom the correspondingc-shift equations in C-2. Our findings have been supported withseveral examples that illustrate these differences. Additionally, the introduction of twonew lemmas not explored in existing literature further adds depth to the mathematicaltools available for addressing such problems.