There is a Neumann theorem, proving that:
$J_0 \big(\sqrt{Z^2+z^2 - 2Zz \cos \phi} \big)=\sum_{k=0}^{\infty} \epsilon_k J_k(Z) J_k (z) \cos k \phi$
where $\epsilon_0=1$, and $\epsilon_k = 2$: $k\geq 1$. It is assumed that $\sqrt{Z^2+z^2 - 2Zz \cos \phi} = |\vec{Z} - \vec{z}|$.
Does anyone know of a generalization for Bessel functions of arbitrary order $J_m \big(\sqrt{Z^2+z^2 - 2Zz \cos \phi} \big)$: $m$ is any integer?