Summation of Powers over Product of Differences/Proof 2

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Theorem

$\ds \sum_{j \mathop = 1}^n \begin{pmatrix} {\dfrac { {x_j}^r} {\ds \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } } \end{pmatrix} = \begin{cases} 0 & : 0 \le r < n - 1 \\ 1 & : r = n - 1 \\ \ds \sum_{j \mathop = 1}^n x_j & : r = n \end{cases}$


Proof

By the Residue Theorem:

$\ds \sum_{j \mathop = 1}^n \begin {pmatrix} {\dfrac { {x_j}^r} {\ds \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } } \end{pmatrix} = \dfrac 1 {2 \pi i} \int \limits_{\size z \mathop = R} \dfrac {z^r \rd z} {\paren {z - z_1} \cdots \paren {z - z_n} }$

where $R > \size {z_1}, \ldots, \size {z_n}$.


The Laurent series of the integrand converges uniformly on $\size z = R$:

\(\ds \) \(\) \(\ds z^{r - n} \paren {\dfrac 1 {1 - x_1 / z} } \cdots \paren {\dfrac 1 {1 - x_n / z} }\)
\(\ds \) \(=\) \(\ds z^{r - n} + \paren {x_1 + \cdots x_n} z^{r - n - 1} + \paren {x_1^2 + x_1 x_2 + \cdots} z^{r - n - 2} + \cdots\)

On integrating term my term, everything vanishes except the coefficient of $z^{-1}$.


Thus:

$\ds \sum_{\substack {j_1 \mathop + \mathop \cdots \mathop + j_n \mathop = r \mathop - n \mathop + 1 \\ j_1, \mathop \ldots j_n \mathop \ge 0} } {x_1}^{j_1} {x_2}^{j_2} \cdots {x_n}^{j_n}$

$\blacksquare$


Historical Note

The result Summation of Powers over Product of Differences was discussed by Leonhard Paul Euler in a letter to Christian Goldbach in $1762$.

He subsequently published it in his Institutiones Calculi Integralis, Volume 2 of $1769$.

The proof involving complex analysis was devised in $1857$ by James Joseph Sylvester.


Sources