there is one useful tool introduced in \[\[BN1\]\] called $\boxplus$-divisibility indicator: $$\mathrm{Ind}(\mu)=\sup\{t\geq0:\mu\in\mathbb{B}_t(\mathcal{M})\},\;\;\;\;\;\mu\in\mathcal{M}.$$ Any measure $\mu\in\mathcal{M}$ with finite support has $\mathrm{Ind}(\mu)=0$, while $\mu$ is $\boxplus$-infinitely divisible if and only if $\mathrm{Ind}(\mu)\geq1$. In general, for any $t\geq0$ and $\mu\in\mathcal{M}$ we have $$\label{Nindicator}
\mathrm{Ind}(\mathbb{B}_t(\mu))=t+\mathrm{Ind}(\mu)$$ For $\boxplus$-divisibility indicators of some specific measures, we refer the reader to \[\[BN1\]\].

$\boxplus$-infinite divisibility of $\mathbb{B}_{p,q}(\mu)$
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For any $\mu\in\mathcal{M}$ and $p\geq1$, $q>0$, denote $\mathbb{B}_{p,q}(\mu)=\left(\mu^{\boxplus p}\right)^{\uplus q}$. Particularly, $\mathbb{B}_{t+1,1/(t+1)}=\mathbb{B}_t$ for any $t\geq0$ and $\mathbb{B}_{2,1/2}=\mathbb{B}$. In this section, we mainly use Proposition \[prop2.1\] to investigate the measure $\mathbb{B}_{p,q}(\mu)$. Throughout the paper, the number $r^*$ stands for the conjugate exponent of any number $r>0$, i.e., $$\frac{1}{r}+\frac{1}{r^*}=1,\;\;\;\;\;r\neq1$$ and $r^*=\infty$ if $r=1$. Note that we have $r^*<0$ if $r\in(0,1)$.

For $p>1$, by (\[p-power\]) and the definition of Boolean convolution power we have $$\label{general}
F_{\mathbb{B}_{p,q}(\mu)}(z)=\frac{pq\omega_p(z)-(1+pq-p)z}{p-1},\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R}.$$ As a special case of (\[general\]), if $1+pq-p=0$, i.e., $q=1/p^*$ then $$F_{\mathbb{B}_{p,1/p^*}(\mu)}(z)
=\frac{p\omega_p(z)}{p^*(p-1)}=\omega_p(z),\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R}.$$ This yields that the Voiculescu transform $\phi_{\mathbb{B}_{p,1/p^*}(\mu)}$ of the measure $\mathbb{B}_{p,1/p^*}(\mu)$ has an analytic continuation to $\mathbb{C}^+$, which is given by $$\begin{aligned}
\phi_{\mathbb{B}_{p,1/p^*}(\mu)}(z)&=F_{\mathbb{B}_{p,1/p^*}(\mu)}^{-1}(z)-z=H_p(z)-z \\
&=E_{\mu^{\uplus(p-1)}}(z),\;\;\;\;\;z\in\mathbb{C}^+.\end{aligned}$$ These observations are recorded in the following result.

\[3.1\] For any measure $\mu\in\mathcal{M}$ and number $p>1$, the measure $\mathbb{B}_{p,1/p^*}(\mu)$ is $\boxplus$-infinitely divisible, the function $F_{\mathbb{B}_{p,1/p^*}(\mu)}$ extends continuously to $\mathbb{C}^+\cup\mathbb{R}$, $$F_{\mu^{\boxplus 