I am trying to simplify this as elegantly as possible:
\(\displaystyle \displaystyle \sum_{k=0}^{n+1}(-1)^{k+n+1}(t_k)^{n+1}\prod_{\substack{i>j\\i,j\neq k}}(t_i-t_j)\)
I am assuming the \(\displaystyle t_i\)'s from \(\displaystyle 0\le j < i \le n+1\) are distinct.
I know that it simplifies to \(\displaystyle \prod_{i>j}(t_i-t_j)\), it is actually a classic result, but I'm not sure how to get there. Even the 3x3 case, while not difficult, requires adding and subtracting rogue terms to get the factorization nicely.
This is just the determinant of the matrix with row entries \(\displaystyle 1, t_k, t_k^2,...,t_k^{n+1}\), some of you may have seen it: http://en.wikipedia.org/wiki/Vandermonde_matrix
\(\displaystyle \displaystyle \sum_{k=0}^{n+1}(-1)^{k+n+1}(t_k)^{n+1}\prod_{\substack{i>j\\i,j\neq k}}(t_i-t_j)\)
I am assuming the \(\displaystyle t_i\)'s from \(\displaystyle 0\le j < i \le n+1\) are distinct.
I know that it simplifies to \(\displaystyle \prod_{i>j}(t_i-t_j)\), it is actually a classic result, but I'm not sure how to get there. Even the 3x3 case, while not difficult, requires adding and subtracting rogue terms to get the factorization nicely.
This is just the determinant of the matrix with row entries \(\displaystyle 1, t_k, t_k^2,...,t_k^{n+1}\), some of you may have seen it: http://en.wikipedia.org/wiki/Vandermonde_matrix
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