Free infinite divisibility for generalized power distributions with free Poisson term

We study free infinite divisibility (FID) for a class which is called generalized power distributions with free Poisson term by using a complex analytic technique and a calculation for the free cumulants and Hankel determinants. In particular, our main result implies that (i) if $X$ follows the free Generalized Inverse Gaussian distribution, then $X^r$ follows an FID distribution when $|r|\ge1$, (ii) if $S$ follows the standard semicircle law and $u\ge 2$, then $(S+u)^r$ follows an FID distribution when $r\le -1$, and (iii) if $B_p$ follows the beta distribution with parameters $p$ and $3/2$, then (a) $B_p^r$ follows an FID distribution when $|r|\ge 1$ and $0<p\le 1/2$, and (b) $B_p^r$ follows an FID distribution when $r\le -1$ and $p>1/2$.