Elementary Symmetric Function/Examples

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Examples of Elementary Symmetric Function

Example: $m = 0$

$e_0 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right) = 1$


Example: $m = 1$

\(\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds x_1 + x_2 + \cdots + x_n\)


Example: $m = 2$

\(\ds e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds x_2 x_3 + \cdots + x_2 x_n\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \cdots\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds x_{n - 1} x_n\)


Example: $m = n$

\(\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds x_1 x_2 \cdots x_n\)


Example: $m > n$

Let $m > n$.

Then:

\(\ds \map {e_m} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds 0\)


Example: Monic polynomial coefficients

Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.

The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are sign factors times an elementary symmetric function:

$\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} + \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$


Example: Recursion

Let $\set {z_1, z_2, \ldots, z_{n + 1} }$ be a set of $n + 1$ values, duplicate values permitted.

Then for $1 \le m \le n$:

$\map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } } = z_{n + 1} \map {e_{m - 1} } {\set {z_1, \ldots, z_n} } + \map {e_m} {\set {z_1, \ldots, z_n} }$