variables are bounded, thus the sequence of moments indeed determines a unique probability measure.

4.  The distribution of the sum $X+Y$ for free random variables $X,Y$ with respective distributions $\mu$ and $\nu$ is called the free convolution of $\mu$ and $\nu$, and is denoted by $\mu\boxplus\nu$.

Free GIG distribution
---------------------

In this paper we are concerned with a specific family of probability measures which we will refer to as free GIG (fGIG) distributions.

The free Generalized Inverse Gaussian (fGIG) distribution is a measure $\mu=\mu(\alpha,\beta,\lambda)$, where $\lambda\in\mathbb{R}$ and $\alpha,\beta>0$ which is compactly supported on the interval $[a,b]$ with the density $$\begin{aligned}
    \mu({{\rm d}}x)=\frac{1}{2\pi}\sqrt{(x-a)(b-x)} \left(\frac{\alpha}{x}+\frac{\beta}{\sqrt{ab}x^2}\right){{\rm d}}x,
    \end{aligned}$$ where $0<a<b$ are the solution of $$\begin{aligned}
\label{eq1}
    1-\lambda+\alpha\sqrt{ab}-\beta\frac{a+b}{2ab}=&0\\
    \label{eq2}
    1+\lambda+\frac{\beta}{\sqrt{ab}}-\alpha\frac{a+b}{2}=&0.
    \end{aligned}$$

Observe that the system of equations for coefficients for fixed $\lambda\in \mathbb{R}$ and $\alpha,\beta>0$ has a unique solution $0<a<b$. We can easily get the following

\[prop:ab\] Let $\lambda\in{\mathbb{R}}$. Given $\alpha,\beta>0$, the system of equations , has a unique solution $(a,b)$ such that $$\label{unique}
0<a<b, \qquad |\lambda| \left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}\right)^2<1.$$ Conversely, given $(a,b)$ satisfying , the set of equations – has a unique solution $(\alpha,\beta)$, which is given by $$\begin{aligned}
&\alpha = \frac{2}{(\sqrt{a}-\sqrt{b})^2}\left( 1 + \lambda \left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}\right)^2 \right) >0, \label{eq3}\\
&\beta = \frac{2 a b}{ (\sqrt{a} 