A Summation of Series Involving Bessel Functions and Order Derivatives of Bessel Functions

In this note, we derive the closed-form expression for the summation of series ${\sum }_{n=0}^{\mathrm{\infty }}n{J}_n(x)\mathrm{\partial }{J}_n\mathrm{/}\mathrm{\partial }n$, 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.