Skew-orthogonal polynomials, differential systems and random matrix theory

TL;DR

This paper investigates skew-orthogonal polynomials related to random matrix ensembles, deriving differential systems and compatibility conditions that reveal their structural properties in the context of orthogonal and symplectic matrices.

Contribution

It introduces a differential-difference-deformation system for skew-orthogonal polynomials associated with general polynomial weights in random matrix theory.

Findings

Subsequences satisfy specific differential-difference equations

Solutions form a fundamental system with compatibility conditions

Rank of vectors matches the degree of the potential in quaternion sense

Abstract

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.