Recursion Property of Elementary Symmetric Function/Examples/Product of n+1 Factors

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:

$x_{n + 1} \map {e_n} {\set {x_1, x_2, \ldots, x_n} } = \map {e_{n + 1} } {\set {x_1, x_2, \ldots, x_n, x_{n + 1} } }$

where $\map {e_n} {\set {x_1, x_2, \ldots, x_n} }$ denotes the elementary symmetric function of degree $n$ on $\set {z_1, \ldots, z_n}$.

$\ds \map {e_m} {\set {z_1, \ldots, z_n, z_{n + 1} } }$

$=$

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

$\ds \leadsto \ \ $

$\ds \map {e_{n + 1} } {\set {z_1, \ldots, z_n, z_{n + 1} } }$

$=$

$\ds z_{n + 1} \map {e_n} {\set {z_1, \ldots, z_n} } + \map {e_{n + 1} } {\set {z_1, \ldots, z_n} }$

setting $m \gets n + 1$

$\ds $

$=$

$\ds z_{n + 1} \map {e_n} {\set {z_1, \ldots, z_n} }$

$\blacksquare$