Delta operators, power series distributions and recursions for compound sums

Abstract

This paper uses some basic elements of the finite operator calculus to develop a new recursive method for calculating the probability distribution of a compound sum. The counting random variable here has a power series distribution of convolution type, which allows us to cover and extend a large number of classical counting distributions. This new approach is introduced in an actuarial framework and leads us to generalize the famous Panjer algorithm whose applications are numerous in collective, operational and credit risk models.