The following series arises in an electrostatics problem for a conducting cylinder: $$ V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2} $$ where $J_i$ is the Bessel function of $i^{th}$ order, and $k_n$ is the location of the $n^{th}$ zero of $J_0$. $V$ can be proven to converge for $z>0$, and from numerical tests, converges conditionally also at $z=0$ except for poles at $\rho=0,2,4,6...$

Is there any kind of analytic or asymptotic expression as a function of $\rho$ in the limit as $z\rightarrow0$, for $\rho>2$ in particular?