Waring’s formula
Let x1,…,xn be n indeterminates. For k≥1, let σk be the kth elementary symmetric polynomials in x1,…,xn, and Sk be the kth power sum defined as
Sk=n∑i=1xki. |
Like the Newton’s formula, the Waring formula is a relation
between σk and Sk:
Sk=∑(-1)(i2+i4+i6+…)(i1+i2+…+in-1)!ki1!i2!⋯in!σi11σi22⋯σinn, |
where the summation is over all n-tuples (i1,…,in)∈ℤn with non-negative components, such that
i1+2i2+…+nin=k. |
In particular, when there are two indeterminates, i.e. n=2, the Waring formula reads
xk1+xk2=⌊k/2⌋∑i=0(-1)ikk-i(k-ii)(x1+x2)k-2i(x1x2)i. |
Title | Waring’s formula |
---|---|
Canonical name | WaringsFormula |
Date of creation | 2013-03-22 15:34:26 |
Last modified on | 2013-03-22 15:34:26 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 9 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11C08 |
Synonym | Waring formula |
Related topic | NewtonGirardFormulaSymmetricPolynomials |