Polynomial traces and elementary symmetric functions in the latent roots of a non-central Wishart matrix

TL;DR

This paper introduces an alternative method using umbral operators to compute expected elementary symmetric functions of non-central Wishart matrix roots, simplifying previous hypergeometric and zonal polynomial approaches.

Contribution

It demonstrates how umbral operators can replace hypergeometric functions and zonal polynomials for calculating symmetric functions in Wishart matrices, providing a novel computational tool.

Findings

Umbral operator effectively computes expected symmetric functions.

The method simplifies calculations by focusing on polynomial traces and cumulants.

Provides an alternative to traditional hypergeometric and zonal polynomial techniques.

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.