Selberg Integrals, Super hypergeometric functions and Applications to $\beta$-Ensembles of Random Matrices

We study a new Selberg-type integral with $n+m$ indeterminates, which turns out to be related to the deformed Calogero-Sutherland systems. We show that the integral satisfies a holonomic system of $n+m$ non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff-Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical $\beta$-ensembles of Random Matrix Theory.