Abstract
A lemma is given first which provides an easy method of expressing the product of an $s$th order Vandermonde type determinant and the $k$th and $l$th $(k, l \geqq 0)$ powers of the $r$th and $h$th $(r, h \leqq s)$ elementary symmetric functions (esf's) respectively as a linear compound of determinants. The lemma extends itself readily to the product of powers of any number of esf's up to the $s$th. Using this lemma and some reduction formulae for certain special types of Vandermonde type determinants, a second lemma has been proved to show that certain formulas for the moments of esf's in $s$ non-null characteristic roots $\lambda_i(0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_s < \infty)$ of a matrix can be easily derived from corresponding formulas for the moments of corresponding esf's in $s$ non-null roots, $\theta_i, (0 < \theta_1 \leqq \cdots \leqq \theta_s < 1)$ of another matrix and vice versa. Illustrations are given explaining both lemmas.
Citation
K. C. Sreedharan Pillai. "On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices." Ann. Math. Statist. 35 (4) 1704 - 1712, December, 1964. https://doi.org/10.1214/aoms/1177700392
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