Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-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 $o(1)$, 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.