Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles

We analyze the eigenvalue density for the Laguerre and Jacobi beta-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms 0(1), in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian beta-ensemble, and Laguerre beta-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.