Generalized renewal sequences and infinitely divisible lattice distributions

We introduce an increasing set of classes Γ_a (0?α?1) of infinitely divisible (i.d.) distributions on {0,1,2,…}, such that Γ₀ is the set of all compound-geometric distributions and Γ₁ 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 Γ₁ and by Steutel [7] for Γ₀. 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 Γ_a'.