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Expressing difference product using Vandermonde determinant.

  1. Feb 25, 2017 #1
    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. jcsd
  3. Feb 25, 2017 #2

    LCKurtz

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  4. Feb 26, 2017 #3
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