A central goal of this mathematical research is the use of newly developed relationships between special functions applied to the evaluation of complex integrals. The primary focus is that of the famous Selberg formula for the integration of symmetric functions multiplied against the Vandermonde determinant, acting as a weight. Work will continue addressing possible extensions of this closed form expression. Initial efforts will be made to establish a formula for the ratios of alternantes to the Selberg symmetric functions. It is expected that they will be given explicitly in terms of tournaments. Work will also be done on beta type distributions and their orthogonal polynomials. In addition, the classification of functions which provide extensions of the q- Dyson theorem of Zeilberger and Bressoud will be sought. It is conjectured that not all such functions will turn out to be symmetric.