and $\eta\to  +\infty$ as $\beta \downarrow0$.

2.  If $|\lambda|<1$ then $\delta\to-\infty$ and $\eta \to \alpha/(1-\lambda^2)$ as $\beta \downarrow0$.

3.  If $\lambda=\pm1$ then $\delta\to-\infty$ and $\eta \to +\infty$ as $\beta \downarrow0$.

Regularity of fGIG distribution under free convolution
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In this section we study in detail regularity properties of the fGIG distribution related to the operation of free additive convolution. In the next theorem we collect all the results proved in this section. The theorem contains several statements about free GIG distributions. Each subsection of the present section proves a part of the theorem.

\[thm:sec3\] The following holds for the free GIG measure $\mu(\alpha,\beta,\lambda)$:

1.  It is freely infinitely divisible for any $\alpha,\beta>0$ and $\lambda\in\mathbb{R}.$

2.  The free Levy measure is of the form $$\label{FLM}
            \tau_{\alpha,\beta,\lambda}({{\rm d}}x)=\max\{\lambda,0\} \delta_{1/\alpha}({{\rm d}}x) + \frac{(1-\delta x) \sqrt{\beta (1-\eta x)}}{\pi x^{3/2} (1-\alpha x)} 1_{(0,1/\eta)}(x)\, {{\rm d}}x.$$

3.  It is free regular with zero drift for all $\alpha,\beta>0$ and $\lambda\in\mathbb{R}$.

4.  It is freely self-decomposable for $\lambda \leq -\frac{B^{\frac{3}{2}}}{A\sqrt{9B-8A}}.$

5.  It is unimodal.

Free infinite divisibility and free Lévy measure
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As we mentioned before, having the operation of free convolution defined, it is natural to study infinite divisibility with respect to $\boxplus$. We say that $\mu$ 