What is the $\mathcal Z$-transform of Bessel function $J_0(\alpha n)$ sequence 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 $|\omega| &lt; \alpha$.  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?
The Taylor expansion for the Bessel function of the first kind and 0th order is 
$$
J_{0}\left ( x \right )= \sum_{m=0}^{\infty }\frac{(-1)^{m}}{(m!)^{^{2}}}\left ( \frac{1}{2}x \right )^{2m}
$$

(see http://en.wikipedia.org/wiki/Bessel_function)

So you can basically approximate this as the Z-transform of a polynomial.