  In this note, we derive the closed-form expression for the summation of
series $\sum_{n=0}^{\infty}nJ_n(x)\partial J_n/\partial n$, which is found in
the calculation of entanglement entropy in 2-d bosonic free field, in terms of
$Y_0$, $J_0$ and an integral involving these two Bessel functions. Further, we
point out the integral can be expressed as a Meijer G function.
