Limit for series of Bessel functions evaluated at zeros

Question

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?

Answer 1

Employing the asymptotics of large zeroes of Bessel functions and the large-argument asymptotics of the Bessel functions, it can be shown that the $n$th term of the series behaves like $$ \frac{1}{\sqrt{2n\rho}}\cos\left(\rho\left(n-\tfrac{1}{4} \right)\pi-\tfrac{\pi}{4}\right)\mathrm{e}^{ -\left(n-\tfrac{1}{4}\right)\pi z} + \mathcal{O}_{z,\rho}\!\left(\frac{1}{n^{3/2}}\right). $$ Thus, in terms of the polylogarithm, $$ V = \frac{\mathrm{e}^{\frac{\pi}{4}z}}{\sqrt {2\rho }}\operatorname{Re} \left[ \mathrm{e}^{ - \frac{\pi}{4}(\rho + 1)\mathrm{i}} \operatorname{Li}_{1/2} \left( \mathrm{e}^{\pi \rho \mathrm{i} - \pi z} \right) \right] + \mathcal{O}_{z,\rho } (1). $$ Yo may proceed from here.