THEORY OF GENERALIZED BESSEL-FUNCTIONS

In this paper we discuss the theory of generalized Bessel functions which are of noticeable importance in the analysis of scattering processes for which the dipole approximation cannot be used. We introduce these functions in their standard form and their modified version. We state the relevant generating functions and Graf-type addition theorems. The usefulness of the results to construct a fast algorithm for their quantitative computation is also devised. We comment on the possibility of getting two-index generalized Bessel functions in e.g. the study of sum rules of the type {Mathematical expression}, where J n is the cylindrical Bessel function of the first kind. The usefulness of the results for problems of practical interest is finally commented on. It is shown that a modified Anger function can be advantageously introduced to get an almost straightforward computation of the Bernstein sum rule in the theory of ion waves. © 1990 Società Italiana di Fisica.