Newton's Identities/Proof 1

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

The tools are Viete's Formulas, symmetric function recursion, telescoping sums and homogeneous functions of degree $k$.

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

Create $n$ equations all with left side 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 $1 \le j \le n$:

{{eqn | r = \sum_{i \mathop = 0}^n \paren {-1}^{n-i} {\map {e_{n-i} } X} {\map {p_{i+r} } X}      | Term $\map {e_{m} X$ is zero for $m \gt n$, affecting case $r \gt 0$. }}

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 right side 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).