On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws

We consider a class of probability measures mu(alpha)(s,r) have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify mu(alpha)(s,2) as a free compound Poisson law with Levy measure a monotone alpha-stable law. This implies the free infinite divisibility of mu(alpha)(s,2). Moreover, when symmetric or positive, mu(alpha)(s,2) has a representation as the free multiplication of a free Poisson law and a monotone alpha-stable law. We also investigate the free infinite divisibility of mu(alpha)(s,r) for r not equal 2. Special cases include the beta distributions B(1 - 1/r, 1 + 1/r) which are freely infinitely divisible if and only if 1 <= r <= 2.