Kanter random variable and positive free stable distributions

We derive the representative Bernstein measure of the density of (X_α)^{−α/(1−α)}, 0 < α < 1, where X_α is a positive stable random variable, as a Fox-H function. Up to a factor, this measure describes the law of some function aα of a uniform random variable U on (0,π). The distribution function of a_1−α(U) is then expressed through a H-function and is used to describe more explicitly the density of the analogue of Xα in the setting of free probability theory. Moreover, this density is shown to converge to a function with infinite mass when α → 0⁺, in contrast to the classical setting where X_α is known to converge weakly to the inverse of an exponential random variable. Analytic evidences of the occurence of a_α in both the classical and the free settings conclude the exposition.