Expressing difference product using Vandermonde determinant.

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1. Feb 25, 2017

1. The problem statement, all variables and given/known data
Show that $g=g(x_1,x_2,...,x_n)=(-1)^{n}V_{n-1}(x)$ where $g(x_i)=\prod_{i<j} (x_i-x_j)$, $x=x_n$ and $V_{n-1}$ is the Vandermonde determinant defined by
$V_{n-1}(x)=\begin{vmatrix} 1 & 1 & ... & 1 & 1 \\ x_1 & x_2 & ... & x_{n-1} & x_n \\ {x_1}^2 & {x_2}^2 & ... & {x_{n-1}}^2 & {x_n}^2 \\ ... & ... & ... & ... & ... \\ {x_1}^{n-1} & {x_2}^{n-1} & ... & {x_{n-1}}^{n-1} & {x_n}^{n-1} \end{vmatrix}$

2. Relevant equations
N\A

3. The attempt at a solution
After expressing the determinant using the sigma notation I attempted to take a common factor to express it in a similar fashion but to no success. Other than that I really don't know how to approach this (I know I shouldn't say this but it is the case) as I never encountered a proof of this kind, and so I would appreciate some help.

2. Feb 25, 2017

LCKurtz

3. Feb 26, 2017