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

TL;DR

This paper derives a closed-form expression for a series involving Bessel functions and their order derivatives, relevant in quantum entanglement entropy calculations, expressing the result in terms of special functions.

Contribution

It provides a novel closed-form solution for a specific Bessel series involving derivatives, connecting it to Meijer G functions for the first time.

Findings

Closed-form expression for the series involving Bessel functions and derivatives.

Representation of the integral as a Meijer G function.

Application relevance to entanglement entropy calculations.

Abstract

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.