Characterization of a class of bivariate distribution functions

Let FX,Y(x,y) be a bivariate distribution function and Pn(x), Qm(y), n, m = 0, 1, 2,..., the orthonormal polynomials of the two marginal distributions FX(x) and FY(y), respectively. Some necessary conditions are derived for the co-efficients cn, n = 0, 1, 2,..., if the conditional expectation E[Pn(X) [short parallel] Y] = cnQn(Y) holds for n = 0, 1, 2,.... Several examples are given to show the application of these necessary conditions.