Extreme eigenvalues of random matrices from Jacobi ensembles

Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $\beta$-Ensembles are derived for matrices of large size in the r\'egime where $\beta > 0$ is arbitrary and one of the model parameters $\alpha_1$ is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases $\beta = 2$ and/or small values of $\alpha_1$, explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented.