§ 1. Introduction

Rather than specializing down from the most general case: distributions on R, we start with the simplest case: distributions on Z+. This case contains most of the essential features of infinite divisibility, but avoides some of the technical problems of the most general situation; the recurrence relations we encounter here, will be integral equations in the R+-case, and the problems concerning zeroes of densities and tail behaviour are more delicate there and on R. The basic tool here is the probability generating function, rather than the Laplace-Stieltjes transform or the characteristic function.