Some recursions for moments of compound distributions

In the present paper, we deduce some recursions for compound distributions. To introduce our technique of deduction, we start with a recursion developed by De Pril [Scandinavian Actuarial Journal (1986) 117] for univariate distributions with counting distribution from the Panjer class. We then extend this recursion to a wider class of counting distributions and discuss some special cases. Finally, we extend the recursions to the case with multivariate severity distribution.