In this article we employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables to the generalized Stieltjes transforms of the distribution functions ; . The random weights are assumed to be cuts of by ordered statistics of independent and identically uniformly distributed random variables on ; . treated the case using the Schwartz distribution theory. We identified fairly large classes of randomly weighted average distributions by their generalized Stieltjes transforms; in particular including the uniform, Wigner and certain power semicircle distributions.