A q-theorem of Pólya using Hurwitz partial fraction method

We establish a q-counterpart of the method of partial fraction developed by Hurwitz-Pólya to investigate the zeros of q-cosine and q-sine transforms, where \(q \in (0,1)\) $q\in (0,1)$ is a fixed number. We prove reality and simplicity of the zeros and give a precise description of their distribution. The conditions imposed on both q and the integrand are less restrictive than previously assumed in the literature. A direct infinite partial fraction expansion is obtained via q-sampling theory.