Free infinite divisibility for generalized power distributions with free Poisson term

摘要

We study free infinite divisibility (FID) for a class of generalizedpower distributions with free Poisson term by using complex analytic methods and free cumulants. In particular, we prove that (i) if X follows the freegeneralized inverse Gaussian distribution, then the distribution of Xris FIDwhen |r| - 1; (ii) if S follows the standard semicircle law and u 2, thenthe distribution of (S + u)ris FID when r ? ?1; (iii) if Bp follows thebeta distribution with parameters p and 3/2, then (iii-a) the distribution ofBrp is FID when |r| - 1 and 0 p ? 1/2; (iii-b) the distribution of Brp isFID when r ? ?1 and p 1/2.