is freely infinitely divisible if for any $n\geq 1$ there exists a probability measure $\mu_n$ such that $$\mu=\underbrace{\mu_n\boxplus\ldots\boxplus\mu_n}_{\text{$n$ times}}.$$ It turns out that free infinite divisibility of compactly supported measures can by described in terms of analytic properties of the $R$-transform. In particular it was proved in [@Voi86 Theorem 4.3] that the free infinite divisibility is equivalent to the inequality ${\text{\normalfont Im}}(r_{\alpha,\beta,\lambda}(z)) \leq0$ for all $z\in{\mathbb{C}}^-$.

As in the classical case, for freely infinitely divisible probability measures, one can represent its free cumulant transform with a Lévy–Khintchine type formula. For a probability measure $\mu$ on ${\mathbb{R}}$, the *free cumulant transform* is defined by $${\mathcal{C}^\boxplus}_\mu(z) = z r_\mu(z).$$ Then $\mu$ is FID if and only if ${\mathcal{C}^\boxplus}_\mu$ can be analytically extended to ${\mathbb{C}}^-$ via the formula $$\label{FLK2}
{\mathcal{C}^\boxplus}_{\mu}(z)=\xi z+\zeta z^2+\int_{{\mathbb{R}}}\left( \frac{1}{1-z x}-1-z x \,1_{[-1,1] }(x) \right) \tau({{\rm d}}x) ,\qquad z\in {\mathbb{C}}^-,$$ where $\xi \in {\mathbb{R}},$ $\zeta\geq 0$ and $\tau$ is a measure on ${\mathbb{R}}$ such that $$\tau (\{0\})=0,\qquad \int_{{\mathbb{R}}}\min \{1,x^2\}\tau ({{\rm d}}x) <\infty.$$ The triplet $(\xi,\zeta,\tau)$ is called the *free characteristic triplet* of $\mu$, and $\tau$ is called the *free Lévy measure* of $\mu$. The formula is called the *free Lévy–Khintchine formula*.

The above form of free Lévy–Khintchine formula was obtained 