Recursions for distributions functions and stop-loss transforms

For any function f on the non-negative integers, we can evaluate the cumulative function rf given by rf(s) = L~=of(x) from the values of f by the recursion r f(8) = r f(8-1) + f(8). Analogously we can use this procedure t times to evaluate the t-th order cumulative function rtf. As an alternative, in the present paper we shall derive recursions for direct evaluation of rt f when f itself satisfies a certain sort of recursion. We shall also derive recursions for the t-th order tails Nf where Af(8) = L~S+l f(x). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for f includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer (1981). We discuss in particular convolutions and compound functions