  Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal
(Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random
Hermitian matrices, are considered. Averages are shown to satisfy duality
relations like {Beta,N,n} <--> {4/Beta,n,N} for all Beta>0, where N and n
respectively denote the number of eigenvalues and products of characteristic
polynomials. At the edge of the spectrum, matrix integrals of the Airy
(Kontsevich) type are obtained. Consequences on the integral representation of
the multiple orthogonal polynomials and the partition function of the formal
one-matrix model are also discussed. Proofs rely on the theory of multivariate
symmetric polynomials, especially Jack polynomials.
