Kapteyn series arising in radiation problems

In discussing radiation from multiple point charges or magnetic dipoles, moving in circles or ellipses, a variety of Kapteyn series of the second kind arises. Some of the series have been known in closed form for a hundred years or more, others appear not to be available to analytic persuasion. This paper shows how 12 such generic series can be developed to produce either closed analytic expressions or integrals that are not analytically tractable. In addition, the method presented here may be of benefit when one has other Kapteyn series of the second kind to consider, thereby providing an additional reason to consider such series anew