The non-local kinetics of the electrons in a cylindrical, axially symmetric hollow cathode discharge is theoretically investigated. A new multi-term method for solving the inhomogeneous Boltzmann equation in cylindrical coordinates has been developed to adequately describe the pronounced anisotropy of the electron velocity distribution function expected in the cathode fall region. This method is based on a tensorial representation of the expansion of the electron velocity distribution function in spherical harmonics. The resultant hierarchy of component equations has been specified to cylindrical geometry with rotational symmetry and axial uniformity. A subset of component equations could be separated which determines the relevant distribution parts needed to analyze the radial behaviour of important electron properties in the hollow cathode. Using measured profiles of the electric field, this subset has numerically been solved for a helium plasma at a discharge current of some mA and a pressure of few torr. Based on this approach in addition to the radial variation of the energy distribution, that of important transport properties and collision rates is studied and the non-local particle and power balance of the electrons are analysed. Some theoretical results are compared with available experimental ones.