Advancements in q-Hankel Transforms Based on Certain Approach of Big q-Bessel Functions and Applications

This study presents a novel variant of a finite q-Hankel transform derived from big q-Bessel functions and investigates its analytical structure, with particular emphasis on the distribution and properties of its zeros. A key focus is placed on the inherent symmetry in the zero distribution of these transforms, which plays a central role in their analytical characterization. We establish rigorous conditions under which the finite q-Hankel transforms exhibit only real zeros and demonstrate their adherence to well-defined asymptotic and symmetric patterns. Moreover, we introduce a series of q-analogs to classical theorems, such as those of Pólya, further illustrating the symmetric nature of these results within the framework of q-calculus. The findings not only deepen the understanding of q-integral transforms and their symmetry properties but also underscore their relevance in the broader context of special functions and mathematical analysis.