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In this note, we derive the closed-form expression for the summation ofseries $\sum_{n=0}^{\infty}nJ_n(x)\partial J_n/\partial n$, which is found inthe 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, wepoint out the integral can be expressed as a Meijer G function.

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