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Examples of the $\beta$-Jacobi ensemble specify the joint distribution of thetransmission eigenvalues in scattering problems. In this context, there hasbeen interest in the distribution of the trace, as the trace corresponds to theconductance. Earlier, in the case $\beta = 1$, the trace statistic was isolatedin studies of covariance matrices in multivariate statistics, where it isreferred to as Pillai's $V$ statistic. In this context, Davis showed that for$\beta = 1$ the trace statistic, and its Fourier-Laplace transform, can becharacterised by $(N+1) \times (N+1)$ matrix differential equations. For theFourier-Laplace transform, this leads to a vector recurrence for the moments.However, for the distribution itself the characterisation provided wasincomplete, as the connection problem of determining the linear combination ofFrobenius type solutions that correspond to the statistic was not solved. Wesolve this connection problem for Jacobi parameter $b$ and Dyson index $\beta$non-negative integers. For the other Jacobi parameter $a$ also a non-negativeinteger, the power series portion of each Frobenius solution terminates to apolynomial, and the matrix differential equation gives a recurrence for theircomputation.

Differential recurrences for the distribution of the trace of the $\beta$-Jacobi ensemble