Generalized renewal sequences and infinitely divisible lattice distributions

We introduce an increasing set of classes [Gamma]a (0[less-than-or-equals, slant][alpha][less-than-or-equals, slant]1) of infinitely divisible (i.d.) distributions on {0,1,2,...}, such that [Gamma]0 is the set of all compound-geometric distributions and [Gamma]1 the set of all compound-Poisson distributions, i.e. the set of all i.d. distributions on the non-negative integers. These classes are defined by recursion relations similar to those introduced by Katti [4] for [Gamma]1 and by Steutel [7] for [Gamma]0. These relations can be regarded as generalizations of those defining the so-called renewal sequences (cf. [5] and [2]). Several properties of i.d. distributions now appear as special cases of properties of the [Gamma]a'.