is $$(1+pq-q)^*=\frac{1+pq-q}{pq-q},$$ by Proposition \[3.1\] we see that $\omega=F_\nu$, where $$\label{nu} \nu=\mathbb{B}_{1+pq-q,\frac{pq-q}{1+pq-q}}(\mu).$$ Then using the definition of the Boolean convolution power and (\[nu\]) gives $$\frac{p\omega(z)-z}{p-1}=\frac{pF_\nu(z)-z}{p-1}=F_{\nu^{\uplus p^*}}(z),\;\;\;\;\;z\in\mathbb{C}^+,$$ and $\nu^{\uplus p^*}=\mathbb{B}_{1+pq-q,\frac{pq}{1+pq-q}}(\mu)$, whence the formula in (\[formula\]) follows. The equality in (\[two\]) follows directly from (\[formula\]). Finally, note that if $t\in[0,1/(p^*q)]$ then $$p(1-qt)\geq1\;\;\;\;\;\mathrm{and}\;\;\;\;\;\frac{q}{1-qt}>0,$$ whence the measure $\mathbb{B}_{p(1-qt),q/(1-qt)}(\mu)$ is defined and (\[t\]) holds by (\[two\]).

If $p>1$ and $0<q<1/p^*$ (or, equivalently, $1+pq-p<0$) then (\[t\]) yields $\mathbb{B}_{p,q}=\mathbb{B}_1\circ\mathbb{B}_{p_1,q_1}$, where $$p_1=p(1-q)>1\;\;\;\;\;\mathrm{and}\;\;\;\;\;q_1=\frac{q}{1-q}>0.$$ Using Proposition \[3.1\] and the preceding discussions gives the following result.

\[3.3\] If $\mu\in\mathcal{M}$, $p>1$, and $0<q\leq1/p^*$ then the following statements hold.

1.  [The measure $\mathbb{B}_{p,q}(\mu)$ is $\boxplus$-infinitely divisible.]{}

2.  [For any $n\in\mathbb{N}$, $${\mathbb{B}_{p,q}(\mu)}=
    \left({\mathbb{B}_{\frac{p(n+q-nq)}{n},\frac{q}{n(1-q)+q}}(\mu)}\right)^{\boxplus
    n}.$$]{}

3.  [The Voiculescu transform of $\mathbb{B}_{p,q}(\mu)$ can be expressed as $$\phi_{\mathbb{B}_{p,q}(\mu)}=E_{\mathbb{B}_{p_1,q_1}(\mu)}.$$]{}

4.  [For $r>0$, let $\nu_r=\mathbb{B}_{p_1,r}(\mu)$. Then $$F_{\mathbb{B}_{p,(1-q)r+q}(\mu)}(z)
    =F_{\nu_r}\left(F_{\mathbb{B}_{p,q}(\mu)}(z)\right),\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R}.$$]{}

Particularly, for any $t\geq1$ the measure $\mathbb{B}_t(\mu)$ is $\boxplus$-infinitely divisible, $$\phi_{\mathbb{B}_t(\mu)}=E_{\mathbb{B}_{t-1}(\mu)},$$ and $F_{\mathbb{B}_t(\mu)}$ is the subordination function of $\mu^{\boxplus(t+1)}$ with respect to $\mu^{\boxplus t}$, that is, $$F_{\mu^{\boxplus(t+1)}}(z)=F_{\mu^{\boxplus t}}(F_{\mathbb{B}_t(\mu)}(z)),
\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R}.$$

The assertions (1) and (3) were proved (particularly, $p=t+1$ and $q=(t+1)^{-1}$ satisfy the condition $1+pq-p\leq0$ if and only if $t\geq1$). Next, observe that $$\frac{p(n+q-nq)}{n}>p-pq\geq1\;\;\;\;\;\mathrm{and}\;\;\;\;\;
\frac{q}{n(1-q)+q}>0,$$ whence the assertion (2) follows from (\[two\]). By (3), $F_{\mathbb{B}_{p,q}(\mu)}^{-1}$ can be expressed as $$\label{1}
F_{\mathbb{B}_{p,q}(\mu)}^{-1}(z)=
\left(1+\frac{q_1}{r}\right)z-\frac{q_1}{r}F_{\nu_r}(z),\;\;\;\;\;z\in\mathbb{C}^+.$$ 