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$$e_t(x_2,\ldots,x_n) = \sum^{n-t}_{i=1} (-1)^{i+1} \frac{e_{t+i}(x_1,\ldots,x_n)}{x^i_1}\text{ for every } 0\leq t < n.$$

$e_t$ is the $t^{th}$ elementary symmetric polynomial in the variabel $\{x_2\ldots x_n\}$
$e_{t+i}$ is the ${(t+i)}^{th}$ elementary symmetric polynomial in the variabel $\{x_1\ldots x_n\}$.

The only proving method i familiar with is induction, in which i stuck at the inductive step.
Any idea what method should i use to prove this? Please give me a lead where to start.

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I would rephrase this as: e_t(x_1, ..,x_n-1) = etc and divide the summand by (x_n)^i instead of singeling out x_1. This keeps the LHS in the familiar form. Remark that, because of symmetry, x_1 and x_n are interchangable. –  Wouter M. Jul 29 at 10:28

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