Free infinite divisibility for powers of random variables

We prove that $X^r$ follows a free regular distribution, i.e. the law of a nonnegative free Lévy process if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r ∈ (−∞, 0] ∪ [1,∞)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r ≥ 1$; (3) $X$ follows a mixture of some HCM distributions and $|r| ≥ 1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r ∈ (−∞, 0]∪[2,∞)$. Also we consider the symmetrization of the above probability measures, and in particular show that $|S|^r$ sign($S$) is freely infinitely divisible for $r ≥ 2$. Therefore $S^n$ is freely infinitely divisible for every $n ∈ N$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables