A Generalized Lerche-Newberger Formula

TL;DR

This paper rigorously proves the Lerche-Newberger formula, extends it to multi-dimensional generalized Bessel functions, and applies these results to physical systems, broadening the formula's theoretical and practical scope.

Contribution

It provides a rigorous proof of the Lerche-Newberger formula and extends it to multi-dimensional generalized Bessel functions with applications to physical systems.

Findings

Proof of the Lerche-Newberger formula

Extension to multi-dimensional generalized Bessel functions

Application to physical systems

Abstract

The Lerche-Newberger formula simplifies harmonic sums of Bessel functions and has seen application in plasma physics and frequency modulated quantum systems. In this paper, we rigorously prove the formula and extend the classical result to a family of multi-dimensional extensions of the single variable Bessel functions called generalized Bessel functions. Since prevailing definitions of these functions do not accommodate arbitrary complex order, we use an auxiliary family of functions called generalized Anger functions and show that the single-variable result holds in multiple dimensions for a certain selection of parameters. We conclude by applying these results to physical systems.