On a special Kapteyn series

Abstract

We investigate the mathematical properties of the function T(ε) =∞n=1 n−1Jn(nε), 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.