Recursion Property of Elementary Symmetric Function/Examples
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Examples of Use of Recursion Property of Elementary Symmetric Function
Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ numbers, duplicate values permitted.
Then for $1 \le m \le n$:
- $\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$
Example: $x_{n + 1} \map {e_n} {\set {x_1, x_2, \ldots, x_n} } = \map {e_{n + 1} } {\set {x_1, x_2, \ldots, x_n, x_{n + 1} } }$
- $x_{n + 1} \map {e_n} {\set {x_1, x_2, \ldots, x_n} } = \map {e_{n + 1} } {\set {x_1, x_2, \ldots, x_n, x_{n + 1} } }$