Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions

Functions characterized by the alternative appelations in the title hereof have application in areas as diverse as heat conduction, probability theory, electronic structure in periodic systems, and hydrology. These functions are regarded as difficult to evaluate and have been given attention by a number of investigators in all these fields. This communication ties together the previously disjoint literature, and presents several new expansions of these functions that are in various parameter ranges computationally more efficient than any of the previously proposed methods of evaluation. A new formula for computation of these functions from nearby tabulated values is also reported. The computational advantages of the new procedures are illustrated with examples.