Newton's Identities/Proof 1

Outline
The proof is divided into three cases: $k < n$, $k = n$ and $k > n$.

The tools are Viète's Formulas, Recursion Property of Elementary Symmetric Function, telescoping sums and homogeneous functions of degree $k$.

Details
Discovery of formulas (1)-(2) has humble beginnings:

Create $n$ equations all with zero by substitution  $x = x_j$ in (3), $1 \le j \le n$.

Then multiply the $j$th equation by $x_j^r$ and add the $n$ equations, for $1 \le j \le n$:

Case $r = 0$ in (4) gives (1) for $k = n$, by isolating term $n \map {e_n} X$.

Case $r > 0$ in (4) gives (2) by re-indexing: $m = k - n + i$ with $m = k - n$ to $k$.

Case $k < n$ in (1), not yet discussed, does not use multiply and add as in (4).

The key ingredient:

The of (1) is:

It remains to prove (7) matches (1):

Define:


 * $\map {y_i} t = t\, x_i, 1 \le i \le n$


 * $X_t = \set {\map {y_1} t, \ldots, \map {y_n} t}$


 * $\map f t = \map {e_k} {tx_1, \ldots, tx_n}$.

Then there are two equations (9)-(10) for $\map {f'} t$:

Let $t = 1$ in (9) and (10) to prove (8).