On the Zeros of $q$-Hankel Transform by Using P\'{o}lya-Hurwitz Partial Fraction Method

TL;DR

This paper applies a modified Pólya-Hurwitz partial fraction method to analyze the zeros of finite $q$-Hankel transforms, relaxing previous restrictions on $q$ and providing new theoretical insights and examples.

Contribution

It introduces a $q$-counterpart of the Pólya-Hurwitz technique that relaxes earlier constraints on $q$, advancing the understanding of zeros of $q$-Hankel transforms.

Findings

New $q$-partial fraction representations derived

Relaxed conditions on $q$ for zero analysis

Experimental examples demonstrating the method

Abstract

The technique of P\'{o}lya-Hurwitz of partial fractions is implemented to investigate the zeros of finite $q$-Hankel transforms, which are defined in terms of the third $q$-Bessel function of Jackson. The new approach, which is a $q$-counterpart of P\'{o}lya-Hurwitz technique relaxes the restrictive conditions imposed on $q$ in the previously obtained results. In the present study, we use the $q$-type sampling theorems of the $q$-Hankel transforms, which lead directly to $q$-partial fractions. Various experimental examples are established.