Generalized Bessel functions and Kapteyn series

Abstract

Kapteyn series are introduced for generalized Bessel functions in full analogy with ordinary Bessel functions. Apart from the relevance that Kapteyn series have in astronomical problems and in electromagnetic radiation theory, interest in this type of series is justified by the possibility of expanding analytic functions of a complex variable in series of Bessel functions with specific features. Accordingly, in this paper, it is shown that it is possible to obtain expansions of m-variable analytic functions in series of GBF's, which reproduce the structure of Kapteyn series of ordinary Bessel functions. The possible field of applications is also suggested.