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