p}}(z)=F_\mu\left(F_{\mathbb{B}_{p,1/p^*}(\mu)}(z)\right),\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R},$$ and the Voiculescu transform of $\mathbb{B}_{p,1/p^*}(\mu)$ can be expressed as $$\phi_{\mathbb{B}_{p,1/p^*}(\mu)}=E_{\mu^{\uplus(p-1)}}.$$ In particular, the above statements hold for $\mathbb{B}_1$.

Observe that Proposition \[3.1\] provides an easy way to prove that $\mathbb{B}_1(\mu)$, $\mu\in\mathcal{M}$, is identically equal to the image $\mathbb{B}(\mu)$ of $\mu$ under the Boolean Bercovici-Pata bijection. Indeed, by Theorem \[thm2.2\] and Proposition \[3.1\] we obtain $$\label{bijection}
\phi_{\mathbb{B}(\mu)}=E_\mu=\phi_{\mathbb{B}_1(\mu)}.$$ In \[\[Nica\]\], results similar to Proposition \[3.1\] for the joint distributions for $k$-tuples of selfadjoint elements in a $C^*$-probability space were obtained by combinatorial tools. We refer the reader to the same paper for the so-called $k$-tuple Boolean Bercovici-Pata bijection and related results.

The following lemma contains some basic properties of the map $\mathbb{B}_{p,q}$ which is frequently used in the sequel. The identity in (\[formula\]) can be obtained by \[\[BN1\], Proposition 3.1\]. Here we provide an alternative proof using Proposition \[3.1\].

If $\mu\in\mathcal{M}$, $p,p_1\geq1$, and $q,q_1>0$ then $$\label{formula}
\left(\mu^{\uplus q}\right)^{\boxplus
p}=\mathbb{B}_{1+pq-q,\frac{pq}{1+pq-q}}(\mu),$$ $$\label{two}
\mathbb{B}_{p_1,q_1}\circ\mathbb{B}_{p,q}=
\mathbb{B}_{p(1+p_1q-q),\frac{p_1q_1q}{1+p_1q-q}},$$ and $$\label{t}
\mathbb{B}_{p,q}=\mathbb{B}_t\circ\mathbb{B}_{p(1-qt),\frac{q}{1-qt}},
\;\;\;\;\;0\leq t\leq\frac{1}{p^*q}.$$

It suffices to show the lemma for $p>1$. By Proposition \[prop2.1\], we have $$F_{(\mu^{\uplus q})^{\boxplus p}}(z)=\frac{p\omega(z)-z}{p-1},\;\;\;\;\;z\in\mathbb{C}^+,$$ where the function $\omega$ is the right inverse of the function $$\begin{aligned}
H(z)&=pz+(1-p)F_{\mu^{\uplus q}} \\
&=(1+pq-q)z+(q-pq)F_\mu(z),\;\;\;\;\;z\in\mathbb{C}^+.\end{aligned}$$ On the other hand, since the number $1+pq-q>1$ whose conjugate exponent 