and such that $Y_1$ and $Y_2$ have Gamma distributions with suitable parameters and we assume that $$\begin{aligned}
X\stackrel{d}{=}\frac{1}{Y_2+\frac{1}{Y_1+X}}\end{aligned}$$ then $X$ necessarily has a GIG distribution. A simpler version of this theorem characterizes smaller class of fGIG distributions by equality $$\begin{aligned}
 \label{eq:char}
X\stackrel{d}{=}\frac{1}{Y_1+X}\end{aligned}$$ for $X$ and $Y_1$ as described above.

The overall result of this paper is that the two distributions GIG and fGIG indeed have many similarities. We show that fGIG distribution is freely infinitely divisible and even more that it is free regular. Moreover fGIG distribution can be characterized by the equality in distribution , where one has to replace the independence assumption by freeness and assume that $Y_1$ has free Poisson distributions. While there are only several examples of freely selfdecomposable distributions it is interesting to ask whether fGIG has this property. It turns out that selfdecomposability is the point where the symmetry between GIG and fGIG partially breaks down: not all fGIG distributions are freely selfdecomposable. We find conditions on the parameters of fGIG family for which this distributions are freely selfdecomposable. Except from the results mentioned above we prove that fGIG distribution is unimodal. We also point out that in [@Fer06] it was proved that fGIG 