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.

翻译：$\beta$- Jacobi 共合体的例子 $\ beta$- Jacobi Commonble 的示例指定了传播源值在散落问题中的联合分布。 在这方面,人们一直对微量分布感兴趣,因为微量与导演相对应。 早些时候, $\beta = 1美元, 在多变量统计中, 微量统计在对共变量矩阵的研究中分离了微量统计, 称之为 Pillai 的 $V$ 统计。 在这方面, Davis 显示, $\beta = 1美元 的跟踪统计和 Fourier- Laplace 变换, 可以用 $ (N+1)\ 时间(N+1) $ 矩阵差异方程式来表示。 对于 Fourier- Laplace 变换, 这会导致瞬间矢量矩阵复现。 然而, 提供的特性本身是不完全的, 因为确定与统计匹配的Frobenbunius 类型解决方案的线性组合的关联问题没有解决。 我们解决了Jacobi 参数参数参数和 Dyson indaxeta exex exmal 等分解法的数值中的非正数模型。