Suppose $j_{\ell,k}$ is the $k$-th zero of Bessel of first kind $J_{\ell}(x)$. I have two summation of form (where $a \leq b$)
$$ \sum_{\ell = - \infty}^{\infty} \sum_{k=1}^{\infty} \frac{J^2_{\ell}(\frac{j_{\ell,k} \times a}{b})}{b^2\times J^2_{\ell+1}(j_{\ell,k})} \exp \left(- \frac{j^{2}_{\ell,k} \times x}{b^2} \right) $$
and
$$ \sum_{\ell = - \infty}^{\infty} \sum_{k=1}^{\infty} \frac{J^2_{\ell+1}(\frac{j_{\ell,k} \times a}{b})}{b^2 \times J^2_{\ell+1}(j_{\ell,k})} \exp \left(- \frac{j^{2}_{\ell,k} \times x}{b^2} \right) $$
I used Mathematica and found the first sum converge to $1/4x$ and the second one to $1/5x$. I am wondering, if there exist a way to prove it mathematically.
Any help is appreciated!