Research Article

On a special Kapteyn series

Published in: Quaestiones Mathematicae
Volume 47 , issue sup1 , 2024 , pages: 213–231

DOI: 10.2989/16073606.2023.2287841

Author(s): A.J.E.M. Janssen Eindhoven University of Technology, The Netherlands

Keywords: Primary: 33C10 , Secondary: 42A16 , Kapteyn series of the first kind , Kepler’s equation , conjugate Fourier series

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Abstract

We investigate the mathematical properties of the function , ϵ ∈ [−1, 1], which is a special Kapteyn series of the first kind. Unlike various other special Kapteyn series, the function T (ϵ) does not seem to possess a closed-form expression. We derive an integral representation for T (ϵ) from which various properties of T (ϵ) can be established. In particular, monotonicity and convexity properties of T (ϵ) and ϵ−1 T (ϵ) can be shown. Also, the behaviour of T (ϵ) as ϵ ↑ 1 can be determined from the integral representation. Furthermore, while the Kapteyn series representation of T (ϵ) is very slowly convergent when ϵ is close to ±1, a regularized form of the integral representation of T (ϵ) allows to compute T (ϵ) accurately using Simpson’s rule with relatively few sample points of the involved integrand.

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