Symbolic methods for the evaluation of sum rules of Bessel functions

TL;DR

This paper employs symbolic methods to simplify the derivation of sum rules for Bessel functions, rederiving known results and discovering new identities relevant to physics applications.

Contribution

The paper introduces a symbolic umbral formalism approach to derive and extend sum rules for Bessel functions, including new closed-form identities.

Findings

Rederived known sum rules and addition theorems for Bessel functions.

Derived new sum rules involving special polynomials and Bessel functions.

Applications demonstrated in plasma physics and quantum optics.

Abstract

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.