Abstract 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.

中文翻译：

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非中心 Wishart 矩阵的潜根中的多项式迹和初等对称函数

摘要 超几何函数和纬向多项式是文献中常用的工具，用于处理非中心 Wishart 潜根中初等对称函数的期望值。这里提出的方法通过使用应用于合适多项式矩阵及其累积量的迹的本影算子来恢复这些对称函数的期望值。de Waal 在 (1972) 中推测使用合适的线性算子代替超几何函数和区域多项式。在这里，我们展示了本影算子如何完成这项任务，并因此代表了处理这些对称函数的替代工具。当特殊形式变量插入变量时，通过本影算子的评估删除了潜在根中的所有单项式，除了那些对基本对称函数有贡献的单项式。累积量利用多项式迹的卷积结构进一步简化了计算。未解决的问题在论文末尾得到解决。