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We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.
Mezzadri, F., Reynolds, A., & Winn, B. (2017). Moments of the eigenvalue densities and of the secular coefficients of β-ensembles. Nonlinearity, 30(3), . https://doi.org/10.1088/1361-6544/aa518c