  Skew orthogonal polynomials arise in the calculation of the $n$-point
distribution function for the eigenvalues of ensembles of random matrices with
orthogonal or symplectic symmetry. In particular, the distribution functions
are completely determined by a certain sum involving the skew orthogonal
polynomials. In the cases that the eigenvalue probability density function
involves a classical weight function, explicit formulas for the skew orthogonal
polynomials are given in terms of related orthogonal polynomials, and the
structure is used to give a closed form expression for the sum. This theory
treates all classical cases on an equal footing, giving formulas applicable at
once to the Hermite, Laguerre and Jacobi cases.
