Skew-orthogonal polynomials, differential systems and random matrix theory

We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.