On a special Kapteyn series

We investigate the mathematical properties of the function (Figure presented.), 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.