PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
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) ]
(view preamble)

View style:

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
Log in to rate this entry.
(view current ratings)

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 2145 times total.

Classification:
AMS MSC: 11C08 (Number theory :: Polynomials and matrices :: Polynomials)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)