Determinants of Random Matrices and Jack Polynomials of Rectangular Shape

We consider an N-dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian N × N matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.