$\mu(\alpha,\beta,\lambda)$. By the Stieltjes inversion formula mentioned in Remark \[rem:Cauchy\], the absolutely continuous part of the free Lévy measure has the density $$-\lim_{{\varepsilon}\to0}\frac{1}{\pi x^2}{\text{\normalfont Im}}(r_{\alpha,\beta,\lambda}(x^{-1}+{{\rm i}}{\varepsilon})), \qquad x \neq 0,$$ atoms are at points $1/p~(p\neq0)$, such that the weight is given by $$\tau_{\alpha,\beta,\lambda}(\{1/p\})=\lim_{z\to p} (p - z) r_{\alpha,\beta,\lambda}(z),$$ is non-zero, where $z$ tends to $p$ non-tangentially from ${\mathbb{C}}^-$. In our case the free Lévy measure does not have a singular continuous part since $r_{\alpha,\beta,\lambda}$ is continuous on ${\mathbb{C}}^-\cup{\mathbb{R}}\setminus\{\alpha\}$. Considering – and we obtain the free Lévy measure $$\tau_{\alpha,\beta,\lambda}({{\rm d}}x)=\max\{\lambda,0\} \delta_{1/\alpha}({{\rm d}}x) + \frac{(1-\delta x) \sqrt{\beta (1-\eta x)}}{\pi x^{3/2} (1-\alpha x)} 1_{(0,1/\eta)}(x)\, {{\rm d}}x.$$ Recall that $\eta \geq \alpha >0>\delta$ holds, and $\eta=\alpha$ if and only if $\lambda=0$. The other two parameters $\xi_{\alpha,\beta,\lambda}$ and $\zeta_{\alpha,\beta,\lambda}$ in the free characteristic triplet will determined in Section \[sec:FR\].

Free regularity {#sec:FR}
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In this subsection we will deal with a property stronger than free infinite divisibility, so called free regularity.\
Let $\mu$ be a FID distribution with the free characteristic triplet $(\xi,\zeta,\tau)$. When the semicircular part $\zeta$ is zero and the free Lévy measure $\tau$ satisfies a stronger integrability property $\int_{{\mathbb{R}}}\min\{1,|x|\}\tau({{\rm d}}x) < \infty$, then the free Lévy-Khintchine representation reduces to $$\label{FR}
{\mathcal{C}^\boxplus}_\mu(z)=\xi' z+\int_{{\mathbb{R}}}\left( 