On the relation between orthogonal, symplectic and unitary matrix ensembles

For the unitary ensembles of N x N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight Function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2x2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w'/w is a rational Function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w'/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.