by Barndorff-Nielsen and Thorbj[ø]{}rnsen [@BNT02b] and it has a probabilistic interpretation (see [@Sat13]). Another form was obtained by Bercovici and Voiculescu [@BV93], which is more suitable for limit theorems.

In order to prove that all fGIG distributions are freely infinitely divisible we will use the following lemma.

\[lem:32\] Let $f\colon (\mathbb C^{-} \cup \mathbb R) \setminus \{x_0\} \to \mathbb C$ be a continuous function, where $x_0\in\mathbb{R}$. Suppose that $f$ is analytic in $\mathbb{C}^{-}$, $f(z)\to 0$ uniformly with $z\to \infty$ and ${\text{\normalfont Im}}(f(x))\leq 0$ for $x \in \mathbb R \setminus \{x_0\}$. Suppose moreover that ${\text{\normalfont Im}}(f(z))\leq 0$ for ${\text{\normalfont Im}}(z)\leq 0$ in neighbourhood of $x_0$ then ${\text{\normalfont Im}}(f(z))\leq 0$ for all $z\in \mathbb{C}^{-}$.

Since $f$ is analytic the function ${\text{\normalfont Im}}f$ is harmonic and thus satisfies the maximum principle. Fix ${\varepsilon}>0$. Since $f(z)\to 0$ uniformly with $z\to \infty,$ let $R>0$ be such that ${\text{\normalfont Im}}f(z)<{\varepsilon}$. Consider a domain $D_{\varepsilon}$ with the boundary $$\partial D_{\varepsilon}=[-R,x_0-{\varepsilon}] \cup \{x_0+{\varepsilon}e^{{{\rm i}}\theta}: \theta \in[-\pi,0]\} \cup[x_0+{\varepsilon},R]\cup\{R e^{{{\rm i}}\theta}: \theta \in[-\pi,0]\}$$ Observe that on $\partial D_{\varepsilon}$ ${\text{\normalfont Im}}f(z)<{\varepsilon}$ by assumptions, and hence by the maximum principle we have ${\text{\normalfont Im}}f(z)<{\varepsilon}$ on whole $D_{\varepsilon}$. Letting ${\varepsilon}\to 0$ we get that ${\text{\normalfont Im}}f(z) \leq 0$ on $\mathbb{C}^{-}$.

Next we proceed 