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Waring's formula (Theorem)

Let $ x_1,\ldots, x_n$ be $ n$ indeterminates. For $ k\geq 1$, let $ \sigma_k$ be the $ k$th elementary symmetric polynomials in $ x_1, \ldots, x_n$, and $ S_k$ be the $ k$th power sum defined as

$\displaystyle S_k = \sum_{i=1}^n x_i^k. $

Like the Newton's formula, the Waring formula is a relation between $ \sigma_k$ and $ S_k$:

$\displaystyle S_k = \sum (-1)^{(i_2+i_4+i_6+\ldots)} \frac{(i_1+i_2+\ldots+i_n-1)!k}{i_1!i_2!\cdots i_n!} \sigma_1^{i_1} \sigma_2^{i_2} \cdots \sigma_n^{i_n}, $
where the summation is over all $ n$-tuples $ (i_1,\ldots, i_n)\in\mathbb{Z}^n$ with non-negative components, such that
$\displaystyle i_1+2i_2+\ldots+ni_n = k. $

In particular, when there are two indeterminates, i.e. $ n=2$, the Waring formula reads

$\displaystyle x_1^k + x_2^k = \sum_{i=0}^{\lfloor k/2 \rfloor} (-1)^i\frac{k}{k-i}\binom{k-i}{i}(x_1+x_2)^{k-2i}(x_1x_2)^i. $



"Waring's formula" is owned by alozano. [ full author list (2) | owner history (1) ]
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See Also: Newton-Girard formula for symmetric polynomials

Other names:  Waring formula
Keywords:  elementary symmetric polynomials

Attachments:
proof of Waring's formula (Proof) by kshum
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Cross-references: components, relation, sum, elementary symmetric polynomials, indeterminates
There are 2 references to this entry.

This is version 6 of Waring's formula, born on 2005-11-10, modified 2006-10-30.
Object id is 7480, canonical name is WaringsFormula.
Accessed 2141 times total.

Classification:
AMS MSC: 11C08 (Number theory :: Polynomials and matrices :: Polynomials)
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