are independent if and only if $X$ has a Generalized Inverse Gaussian (GIG) distribution and $Y$ has a Gamma distribution. In the free version of this theorem (i.e. the theorem where one replaces classical independence assumptions by free independence) it turns out that the role of the Gamma distribution is taken again by the free Poisson distribution and the role of the GIG distribution plays a probability measure which appeared for the first time in [@Fer06]. We will refer to this measure as the free Generalized Inverse Gaussian distribution or fGIG for short. We give the definition of this distribution in Section 2.

The main motivation of this paper is to study further properties of fGIG distribution. The results from [@Szp17] suggest that in some sense (but not by means of the BP bijection) this distribution is the free probability analogue of the classical GIG distribution. It is natural then to ask if fGIG distribution shares more properties with its classical counterpart. It is known that the classical GIG distribution is infinitely divisible (see [@BNH77]) and selfdecomposable (see [@Hal79; @SS79]). In [@LS83] the GIG distribution was characterized in terms of an equality in distribution, namely if we take $X,Y_1,Y_2$ independent 