Linear differential equations for the resolvents of the classical matrix ensembles

Abstract

The spectral density for random matrix [Formula: see text] ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of [Formula: see text], which for even [Formula: see text] is a polynomial of degree [Formula: see text]. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree [Formula: see text]. This equation, and its companion for the resolvent, are given explicitly for [Formula: see text] and [Formula: see text] for all three classical cases, and also for [Formula: see text] in the Gaussian case. Known dualities for the spectral moments relating [Formula: see text] to [Formula: see text] then imply corresponding differential equations in the case [Formula: see text], and for the Gaussian ensemble, the case [Formula: see text]. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their [Formula: see text] expansions, along with first-order differential equations for the coefficients of the [Formula: see text] expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.