What is the $\mathcal Z$-transform of the sequence $J_0(\alpha n)$ for $n \in \mathbb{Z}$?
The Fourier transform of zero$^{\rm th}$ order Bessel function $J_0(\alpha x)$ is known to be $\frac{2}{\sqrt{\alpha^2 - \omega^2}}$ for $|\omega| < \alpha$. This has a pole at $\omega = \alpha$. Does this imply that the $\mathcal Z$-transform will also have a pole on the unit circle?
EDIT:
The problem I'm looking at involves discrete samples of Bessel function i.e. $J_0(n)$. How should I proceed to determine its $\mathcal Z$-transform?