- \sqrt{b})^2}\left( 1 - \lambda \left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}\right)^2 \right)>0. \label{eq4}\end{aligned}$$

Thus we may parametrize fGIG distribution using parameters $(a,b,\lambda)$ satisfying instead of $(\alpha,\beta,\lambda)$. We will make it clear whenever we will use a parametrization different than $(\alpha,\beta,\lambda)$.

It is useful to introduce another parameterization to describe the distribution $\mu(\alpha,\beta,\lambda)$. Define $$\label{eq:AB}
    A=(\sqrt{b}-\sqrt{a})^2, \qquad B= (\sqrt{a}+\sqrt{b})^2,$$ observe that we have then $$\begin{aligned}
    \alpha =& \frac{2}{A}\left( 1 + \lambda \frac{A}{B} \right) >0, \qquad
    \beta = \frac{(B-A)^2}{8 A}\left( 1 - \lambda \frac{A}{B} \right)>0,\\
    a =& \left(\frac{\sqrt{B}-\sqrt{A}}{2}\right)^2,\qquad
    b = \left(\frac{\sqrt{A}+\sqrt{B}}{2}\right)^2.
    \end{aligned}$$ The condition is equivalent to $$\label{eq:ABineq}
    0<\max\{1,|\lambda|\}A<B.$$ Thus one can describe any measure $\mu(\alpha,\beta,\lambda)$ in terms of $\lambda,A,B$.

$R$-transform of fGIG distribution {#sec:form}
----------------------------------

The $R$-transform of the measure $\mu(\alpha,\beta,\lambda)$ was calculated in [@Szp17]. Since the $R$-transform will play a crucial role in the paper we devote this section for a detailed study of its properties. We also point out some properties of fGIG distribution which are derived from properties of the $R$-transform.

Before we present the $R$-transform of fGIG distribution let us briefly recall how the $R$-transform is defined and stress its importance for free probability.

\[rem:Cauchy\]

1.  For a probability measure $\mu$ one defines its Cauchy transform via $$G_\mu(z)=\int \frac{1}{z-x}{{\rm d}}\mu(x).$$ It is 