Differential recurrences for the distribution of the trace of the β-Jacobi ensemble

Examples of the β-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scattering problems. For this application, the trace is of relevance as determining the conductance. Earlier, in the case β=1, the trace statistic was isolated in studies of covariance matrices in multivariate statistics. There, Davis showed that for β=1 the trace statistic could be characterised by (N+1)×(N+1) matrix differential equations, now understood for general β>0 as part of the theory of Selberg correlation integrals. However the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameters b and Dyson index β non-negative integers. The distribution then has the functional form of a series of piecewise power functions times a polynomial, and our characterisation gives a recurrence for the computation of the polynomials. For all β>0 we express the Fourier–Laplace transform of the trace statistic in terms of a generalised hypergeometric function based on Jack polynomials.