  Examples of the $\beta$-Jacobi ensemble specify the joint distribution of the
transmission eigenvalues in scattering problems. In this context, there has
been interest in the distribution of the trace, as the trace corresponds to the
conductance. Earlier, in the case $\beta = 1$, the trace statistic was isolated
in studies of covariance matrices in multivariate statistics, where it is
referred to as Pillai's $V$ statistic. In this context, Davis showed that for
$\beta = 1$ the trace statistic, and its Fourier-Laplace transform, can be
characterised by $(N+1) \times (N+1)$ matrix differential equations. For the
Fourier-Laplace transform, this leads to a vector recurrence for the moments.
However, for the distribution itself 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 parameter $b$ and Dyson index $\beta$
non-negative integers. For the other Jacobi parameter $a$ also a non-negative
integer, the power series portion of each Frobenius solution terminates to a
polynomial, and the matrix differential equation gives a recurrence for their
computation.
