Generalized Bessel functions and Kapteyn series (Q1608447)

Kapteyn series are defined by \[ \sum^\infty_{n=0} a_nJ_{\nu+n} \bigl\{(\nu+n) z\bigr\}, \] where \(J_\mu\) is the Bessel function of the first kind and order \(\mu\). In this paper, the authors investigate the possibility of generalizing these series to the multivariable generalized Bessel functions. Then they obtain expansions for some analytic functions of \(m\) variables in terms of these generalized Kapteyn series. The possible field of applications is also suggested.