deals mainly with properties of free GIG distribution related to free probability and in particular to free convolution. Therefore in this section we introduce notions and tools that we need in this paper. The introduction is far from being detailed, reader not familiar with free probability may find a very good introduction to the theory in [@VDN92; @NS06; @MS].

1.  A $C^*$–probability space is a pair $(\mathcal{A},\varphi)$, where $\mathcal{A}$ is a unital $C^*$-algebra and $\varphi$ is a linear functional $\varphi\colon\mathcal{A}\to\mathbb{C}$, such that $\varphi(\mathit{1}_\mathcal{A})=1$ and $\varphi(aa^*)\geq 0$. Here by $\mathit{1}_\mathcal{A}$ we understand the unit of $\mathcal{A}$.

2.  Let $I$ be an index set. A family of subalgebras $\left(\mathcal{A}_i\right)_{i\in I}$ are called free if $\varphi(X_1\cdots X_n)=0$ whenever $a_i\in \mathcal{A}_{j_i}$, $j_1\neq j_2\neq \ldots \neq j_n$ and $\varphi(X_i)=0$ for all $i=1,\ldots,n$ and $n=1,2,\ldots$. Similarly, self-adjoint random variables $X,\,Y\in\mathcal{A}$ are free (freely independent) when subalgebras generated by $(X,\,\mathit{1}_\mathcal{A})$ and $(Y,\,\mathit{1}_\mathcal{A})$ are freely independent.

3.  The distribution of a self-adjoint random variable is identified via moments, that is for a random variable $X$ we say that a probability measure $\mu$ is the distribution of $X$ if $$\varphi(X^n)=\int t^n\,{{\rm d}}\mu(t),\,\mbox{for all } n=1,2,\ldots$$ Note that since we assume that our algebra $\mathcal{A}$ is a $C^*$–algebra, all random 