Recursion Property of Elementary Symmetric Function/Proof 1
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Theorem
Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ numbers, duplicate values permitted.
Then for $1 \le m \le n$:
- $\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$
Proof
Case $m = 1$ holds because $e_0$ is $1$ and $e_1$ is the sum of the elements.
Assume $2 \le m \le n$.
Define four sets:
- $A = \set {\set {p_1, \ldots, p_m} : 1 \le p_1 < \cdots < p_m \le n + 1}$
- $B = \set {\set {p_1, \ldots, p_m} : 1 \le p_1 < \cdots < p_{m - 1} \le n, p_m = n + 1}$
- $C = \set {\set {p_1, \ldots, p_m} : 1 \le p_1 < \cdots < p_m \le n}$
- $D = \set {\set {p_1, \ldots, p_{m - 1} } : 1 \le p_1 < \cdots < p_{m - 1} \le n}$
Then $A = B \cup C$ and $B \cap C = \O$ implies:
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- $\ds \sum_A z_{p_1} \cdots z_{p_m} = \sum_B z_{p_1} \cdots z_{p_m} + \sum_C z_{p_1} \cdots z_{p_m}$
Simplify:
- $\ds \sum_B z_{p_1} \cdots z_{p_m} = z_{n + 1} \sum_D z_{p_1} \cdots z_{p_{m - 1} }$
By definition of elementary symmetric function:
\(\ds \map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } }\) | \(=\) | \(\ds \sum_A z_{p_1} \cdots z_{p_m}\) | ||||||||||||
\(\ds \sum_D z_{p_1} \cdots z_{p_{m - 1} }\) | \(=\) | \(\ds \map {e_{m - 1} } {\set {z_1, \ldots, z_n} }\) | ||||||||||||
\(\ds \sum_C z_{p_1} \cdots z_{p_m}\) | \(=\) | \(\ds \map {e_m} {\set {z_1, \ldots, z_n} }\) |
Assemble the preceding equations:
\(\ds \map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } }\) | \(=\) | \(\ds \sum_A z_{p_1} \cdots z_{p_m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_B z_{p_1} \cdots z_{p_m} + \sum_C z_{p_1} \cdots z_{p_m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z_{n + 1} \sum_D z_{p_1} \cdots z_{p_m} + \sum_C z_{p_1} \cdots z_{p_m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }\) |
$\blacksquare$