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
L\'{e}vy 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 rî€ =2. Special cases
include the beta distributions B(1−r1​,1+r1​) which are
freely infinitely divisible if and only if 1≤r≤2.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ473 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm