  The Jacobi ensemble is one of the classical ensembles of random matrix
theory. Prominent in applications are properties of the eigenvalues at the
spectrum edge, specifically the distribution of the largest (e.g. Roy's largest
root test in multivariate statistics) and smallest (e.g.~condition numbers of
linear systems) eigenvalues. We identify three ranges of parameter values for
which the gap probability determining these distributions is a finite sum with
respect to particular bases, and moreover make use of a certain
differential-difference system fundamental in the theory of the Selberg
integral to provide a recursive scheme to compute the corresponding
coefficients.
