Let X, Y, B be three independent random variables such that X has the same distribution function as Y B. Assume that B is a beta random variable with positive parameters alpha, beta and Y has distribution function H with H(0) = 0. In this paper we derive a recursive formula for calculation of H, if the distribution function H(alpha,beta) of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H(alpha,beta), respectively, and the conditional limiting distribution of bivariate elliptical distributions.