distribution with respect to tensor, free, Boolean and monotone convolutions. Moreover, $\mu_{a,b}$ is a strictly 1-stable distribution with respect to the four convolutions [@Be-Vo; @K-G; @Has1; @S-W]. These facts are proved by calculating the Fourier transform and Stieltjes transform. The calculation for the tensor convolution differs from the other three: it is characterized by the multiplication of Fourier transforms while the other three convolutions are characterized by using Stieltjes transforms (see (\[ast\])-(\[rhd\])). In addition, the Fourier transform strongly differs from the Stieltjes transform: the former is not (real) analytic in general but the latter is always analytic in the complex plane except for the real line. In spite of this difference, the four concepts of infinite divisibility have one-to-one correspondence to each other by using the so-called Bercovici-Pata bijections; the reader is referred to [@Be-Pa; @Has0] and also [@Fra1]. A universal role of Cauchy distributions comes from the fact that they are fixed by the Bercovici-Pata bijections.

We note that there are other nontrivial relations involving these four convolutions. For instance, some free $\frac{1}{2}$-stable distributions are infinitely divisible in the tensor sense [@S-P] (this is also the case in the monotone case, while its explicit calculation is not in the 