Rational characteristic functions and geometric infinite divisibility

Motivated by the fact that exponential and Laplace distributions have rational characteristic functions and are both geometric infinitely divisible (GID), we investigate the latter property in the context of more general probability distributions on the real line with rational characteristic functions of the form P(t)/Q(t), where P(t)=1+a1it+a22(it) and Q(t)=1+b1it+b22(it). Our results provide a complete characterization of the class of characteristic functions of this form, and include a description of their GID subclass. In particular, we obtain characteristic functions in the class and the subclass that are neither exponential nor Laplace.