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

We consider a class of probability measures μs,rα which 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 μs,2α as a free compound Poisson law with Lévy measure a monotone α-stable law. This implies the free infinite divisibility of μs,2α. Moreover, when symmetric or positive, μs,2α has a representation as the free multiplication of a free Poisson law and a monotone α-stable law. We also investigate the free infinite divisibility of μs,rα for r≠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.