$$\label{Taylor}
\sqrt{f_{\alpha,\beta,\lambda}(z)}=\alpha|\lambda| + \frac{\sqrt{\beta}(2\eta-3\alpha+\delta)}{\sqrt{\eta-\alpha}}(z-\alpha) + o(|z-\alpha|),\qquad \mbox{as }z\to \alpha.$$ This implies and and so $r_{\alpha,\beta,\lambda}$ may be extended to a continued function on ${\mathbb{C}}^-\cup {\mathbb{R}}$.

The case $\lambda=0$ follows from the fact that in this case we have $\eta=\alpha$.

In the case $\lambda<0$ one can extend $r_{\alpha,\beta,\lambda}$ to an analytic function in ${\mathbb{C}}^-$ and continuous in ${\mathbb{C}}^- \cup{\mathbb{R}}$.

Some properties of fGIG distribution
------------------------------------

We study here further properties of free GIG distribution. Some of them motivate Section 4 where we will characterize fGIG distribution in a way analogous to classical GIG distribution.

The next remark recalls the definition and some basic facts about free Poisson distribution, which will play an important role in this paper.

\[rem:freePoisson\]

1.  Marchenko–Pastur (or free-Poisson) distribution $\nu=\nu(\gamma, \lambda)$ is defined by the formula $$\begin{aligned}
        \nu=\max\{0,\,1-\lambda\}\,\delta_0+\tilde{\nu},
        \end{aligned}$$ where $\gamma,\lambda> 0$ and the measure $\tilde{\nu}$, supported on the interval $(\gamma(1-\sqrt{\lambda})^2,\,\gamma(1+\sqrt{\lambda})^2)$, has the density (with respect to the Lebesgue measure) $$\tilde{\nu}({{\rm d}}x)=\frac{1}{2\pi\gamma x}\,\sqrt{4\lambda\gamma^2-(x-\gamma(1+\lambda))^2}\,{{\rm d}}x.$$

2.  The $R$-transform of the free Poisson distribution $\nu(\gamma,\lambda)$ is of the form $$r_{\nu(\gamma, \lambda)}(z)=\frac{\gamma\lambda}{1-\gamma z}.$$

The next proposition was proved in [@Szp17 Remark 2.1] which is the free counterpart of a convolution property of classical Gamma and GIG distribution. The proof is 