Methods for Summing General Kapteyn Series

TL;DR

This paper reviews methods for deriving closed-form expressions of Kapteyn series involving Bessel functions, highlighting their applications in physics and mathematics, and demonstrating how they can be expressed as trigonometric or gamma function series.

Contribution

It introduces new approaches to express Kapteyn series in closed form as trigonometric or gamma function series, expanding analytical tools for their evaluation.

Findings

Kapteyn series can be expressed as trigonometric or gamma function series.

Closed-form expressions are achievable for specific parameters.

Examples demonstrate the complexity and applicability of the methods.

Abstract

The general features and characteristics of Kapteyn series, which are a special type of series involving Bessel function, are investigated. For many applications to physics, astrophysics, and mathematics, it is crucial to have closed-form expressions in order to determine their functional structure and parametric behavior. Closed-form expressions of Kapteyn series have mostly been limited to special cases, even though there are often similarities in the approaches used to reduce the series to analytically tractable forms. The goal of this paper is to review the previous work in the area and to show that Kapteyn series can be expressed as trigonometric or gamma function series, which can be evaluated in closed form for specific parameters. Two examples with a similar structure are given, showing the complexity of Kapteyn series.