论文标题
差异复发，用于分布$β$ -JACOBI合奏的痕迹
Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble
论文作者
论文摘要
$β$ -JACOBI合奏的示例指定了散射问题中传输特征值的联合分布。在这种情况下，由于迹线对应于电导率，因此对迹线的分布引起了兴趣。早些时候，在$β= 1 $的情况下，痕量统计量是在多元统计中的协方差矩阵研究中隔离的，在该研究中，它被称为Pillai的$ V $统计量。在这种情况下，戴维斯（Davis）表明，对于$β= 1 $，痕量统计量及其傅立叶宽度变换，可以以$（n+1）\ times（n+1）$矩阵微分方程为特征。对于傅立叶宽带变换，这导致了瞬间的矢量复发。但是，对于分布本身，所提供的表征是不完整的，因为确定与统计量相对应的Frobenius类型解决方案的线性组合的连接问题无法解决。我们解决了Jacobi参数$ B $和Dyson Index $β$非负整数的此连接问题。对于其他Jacobi参数$ A $也是一个非负整数，每个Frobenius解决方案的功率系列部分终止于多项式，并且矩阵微分方程为其计算提供了复发。
Examples of the $β$-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 $β= 1$, 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 $β= 1$ 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 $β$ 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.
