\frac{1}{1-z x}-1\right) \tau({{\rm d}}x) ,\qquad z\in {\mathbb{C}}^-,$$ where $\xi' =\xi -\int_{[-1,1]}x \,\tau({{\rm d}}x) \in {\mathbb{R}}$ is called a *drift*. The distribution $\mu$ is said to be *free regular* [@PAS12] if $\xi'\geq0$ and $\tau$ is supported on $(0,\infty)$. A probability measure $\mu$ on ${\mathbb{R}}$ is free regular if and only if the free convolution power $\mu^{\boxplus t}$ is supported on $[0,\infty)$ for every $t>0$, see [@AHS13]. Examples of free regular distributions include positive free stable distributions, free Poisson distributions and powers of free Poisson distributions [@Has16]. A general criterion in [@AHS13 Theorem 4.6] shows that some boolean stable distributions [@AH14] and many probability distributions [@AHS13; @Has14; @AH16] are free regular. A recent result of Ejsmont and Lehner [@EL Proposition 4.13] provides a wide class of examples: given a nonnegative definite complex matrix $\{a_{ij}\}_{i,j=1}^n$ and free selfadjoint elements $X_1,\dots, X_n$ which have symmetric FID distributions, the polynomial $\sum_{i,j=1}^n a_{ij} X_i X_j$ has a free regular distribution with zero drift.

For the free GIG distributions, the semicircular part can be found by $\displaystyle\zeta_{\alpha,\beta,\lambda}= \lim_{z\to \infty} z^{-1} r_{\alpha,\beta,\lambda}(z)=0$. The free Lévy measure satisfies $${{\rm supp}}(\tau_{\alpha,\beta,\lambda}) \subset (0,\infty), \qquad \int_0^\infty \min\{1,x\} \tau_{\alpha,\beta,\lambda}({{\rm d}}x)<\infty$$ and so we have the reduced formula . The drift is given 