Methods for summing general Kapteyn series

The general features and characteristics of Kapteyn series, which are a special type of series involving the Bessel function, are investigated. For many applications in physics, astrophysics and mathematics, it is crucial to have closed-form expressions in order to determine their functional structu...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2011-09, Vol.44 (38), p.385202-14
Main Authors: Tautz, R C, Lerche, I, Dominici, D
Format: Article
Language:English
Subjects:
Analogies
Astrophysics
Bessel functions
Complexity
Exact sciences and technology
Exact solutions
Gamma function
Mathematical analysis
Physics
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Summary:The general features and characteristics of Kapteyn series, which are a special type of series involving the Bessel function, are investigated. For many applications in physics, astrophysics and mathematics, it is crucial to have closed-form expressions in order to determine their functional structure and parametric behavior. The 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 a closed form for specific parameters. Two examples with a similar structure are given, showing the complexity of Kapteyn series.
ISSN:1751-8121
1751-8113
1751-8121
DOI:10.1088/1751-8113/44/38/385202