Some criteria of rational-infinite divisibility for probability laws

We study the class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. The class $\boldsymbol{Q}$ 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 $\boldsymbol{Q}$ 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.