literature). Gaussian distributions are also such examples: they are infinitely divisible both in the tensor and free senses [@B]. These measures are however not fixed points of the Bercovici-Pata bijection.

Now we go on to the idea of this paper. The Fourier transform of a probability distribution $\mu$ on the real line is defined by $$\mathcal{F}_\mu (z) = \int_{\operatorname*{\mathbb{R}}} e^{ixz}\mu(dx),~~~z\in \operatorname*{\mathbb{R}},$$ and the Stieltjes transform is defined by $$G_\mu(z) = \int_{\operatorname*{\mathbb{R}}} \frac{\mu(dx)}{z - x},~~~ z \in \operatorname*{\mathbb{C}}\backslash \operatorname*{\mathbb{R}}.$$ The reciprocal $F_\mu(z)=\frac{1}{G_\mu(z)}$ is also important in noncommutative probability theory. We use the usual notation for convolutions: $\ast$ for the tensor convolution (the usual convolution), $\boxplus$ for the free convolution, $\uplus$ for the Boolean convolution and $\rhd$ for the monotone convolution. These convolutions are characterized by [@Be-Vo; @Mur3; @V2; @S-W] $$\begin{aligned}
&\mathcal{F}_{\mu_1 \ast \mu_2}(z) = \mathcal{F}_{\mu_1}(z) \mathcal{F}_{\mu_2}(z), ~~z \in \operatorname*{\mathbb{R}}, \label{ast}\\
&F_{\mu_1 \boxplus \mu_2} ^{-1} (z) = F_{\mu_1}^{-1}(z) + F_{\mu_2}^{-1}(z) - z, ~~ z \in \Omega_1 \cap \Omega_2, \label{boxplus}\\ 
&F_{\mu_1 \uplus \mu_2}(z) = F_{\mu_1}(z) + F_{\mu_2}(z) - z, ~~ z \in {\mathbb{C}_{-}}, \label{uplus}\\
&F_{\mu_1 \rhd\mu_2}(z) = F_{\mu_1} (F_{\mu_2}(z)),  ~~z \in {\mathbb{C}_{-}}, \label{rhd}\end{aligned}$$ for probability measures $\mu_1,~\mu_2$. $F_{\mu_k}^{-1}$ is defined in a domain $\Omega_k$ of ${\mathbb{C}_{-}}=\{z \in \operatorname*{\mathbb{C}}; {\text{\normalfont Im~}}z < 0 \}$ 