Frobenius & Special Functions · Superchallenge problem

The Frobenius solutions of Bessel's equation of order $\nu = \tfrac13$ are $J_{\pm\nu}(x) = \displaystyle\sum_{k\ge0}\frac{(-1)^k}{k!\,\Gamma(k \pm \nu + 1)}\left(\frac x2\right)^{2k\pm\nu}$. Compute their Wronskian $W = J_{\nu}J_{-\nu}' - J_{-\nu}J_{\nu}'$ at $x = 1$.