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Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles

The spectral density for random matrix $β$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $β$, which for even $β$ is a polynomial of degree $β(N-1)$. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover the spectral density itself, can be characterised as the solution of a linear differential equation of degree $β+1$. This equation, and its companion for the resolvent, are given explicitly for $β=2$ and $4$ for all three classical cases, and also for $β=6$ in the Gaussian case. Known dualities for the spectral moments relating $β$ to $4/β$ then imply corresponding differential equations in the case $β=1$, and for the Gaussian ensemble, the case $β=2/3$. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their $1/N$ expansions, along with first-order differential equations for the coefficients of the $1/N$ expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.