# Elementary Symmetric Function/Examples

## 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 $1 \le m \le n$:

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