Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials

We show that skew-orthogonal functions, defined with respect to Jacobi weight w(a,b)(x) = (1 - x)(a) (1 + x)(b), a, b > -1, including the limiting cases of Laguerre (w(a) (x) = x(a) e(-x), a > - 1) and Gaussian weight (w(x) = e(-x2)), satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2N x 2N matrices. Using the GCD formulae we calculate the level densities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skew-orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices.