Let $X$ and $Y$ be two unbounded random variables. Then two necessary conditions are proved concerning the structure of the bivariate distribution function of $X$ and $Y$ when it is expanded in the orthonormal polynomials of its marginal distributions. The first condition concerns the shrinking of the polynomial representation into a diagonal form, and the second is a generalization of the Sarmanov-Bratoeva theorem.
"Two Necessary Conditions on the Representation of Bivariate Distributions by Polynomials." Ann. Statist. 4 (1) 216 - 222, January, 1976. https://doi.org/10.1214/aos/1176343355