Journal of Statistical Physics, Vol. 94, Nos. 3/4, 1999. p. 347-363.
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 2 x 2 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.