Some criteria of rational-infinite divisibility for probability laws

We study the class Q of distribution functions F that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions F1 and F2 such that F1=F∗F2. The class Q is a wide natural extension of the fundamental class of infinitely divisible distribution functions. We are interested in general conditions to belong to the class Q in terms of characteristic functions. We obtain criteria that seem to be convenient for the application for some cases, and we illustrate it by several examples in the paper.