论文标题
多项式轨迹和基本对称函数在非中央Wishart矩阵的潜根中
Polynomial traces and elementary symmetric functions in the latent roots of a non-central Wishart matrix
论文作者
论文摘要
高几何函数和区域多项式是文献中通常介绍的工具，以处理非中央Wishart潜在根中基本对称函数的预期值。此处提出的方法通过使用应用于合适的多项式矩阵及其累积物的轨迹的遮罩算子来恢复这些对称函数的期望值。 1972年，德瓦尔（De Waal）猜想了合适的线性运算符代替高几何函数和区域多项式。在这里，我们展示了Umbral Operator如何完成此任务，因此代表了处理这些对称功能的替代工具。当特殊的形式变量被插入变量时，通过Umbral Operator的评估删除了潜在根中的所有单元，除了在基本对称函数中贡献的根源。累积物进一步简化了利用多项式迹线的卷积结构的优势。在论文的末尾解决了开放问题。
Hypergeometric functions and zonal polynomials are the tools usually addressed in the literature to deal with the expected value of the elementary symmetric functions in non-central Wishart latent roots. The method here proposed recovers the expected value of these symmetric functions by using the umbral operator applied to the trace of suitable polynomial matrices and their cumulants. The employment of a suitable linear operator in place of hypergeometric functions and zonal polynomials was conjectured by de Waal in 1972. Here we show how the umbral operator accomplishes this task and consequently represents an alternative tool to deal with these symmetric functions. When special formal variables are plugged in the variables, the evaluation through the umbral operator deletes all the monomials in the latent roots except those contributing in the elementary symmetric functions. Cumulants further simplify the computations taking advantage of the convolution structure of the polynomial trace. Open problems are addressed at the end of the paper.
