PlanetMath: proof of Waring's formula

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proof of Waring's formula

(Proof)

The following is a proof of the Waring's formula using formal power series. We will work with formal power series in indeterminate with coefficients in the ring $\mathbb{Q}[x_1,\ldots,x_n]$ . We also need the following equality

$\displaystyle -\log(1-z) = \sum_{j=1}^\infty \frac{z^j}{j}.$

Taking log on both sides of

$\displaystyle 1 - \sigma_1z+\ldots + (-1)^n \sigma_n z^n = \prod_{m=1}^n(1-x_mz),$

we get

$\displaystyle \log(1 - \sigma_1z+\ldots + (-1)^n \sigma_n z^n) = \sum_{m=1}^n \log(1-x_mz),$

(1)

Waring's formula will follow by comparing the coefficients on both sides.

The right hand side of the above equation equals

$\displaystyle \sum_{m=1}^n \sum_{j=1}^\infty \frac{x_m^j}{j}z^j$

$\displaystyle \sum_{j=1}^\infty \left( \sum_{m=1}^n x_m^j \right) \frac{z^j}{j}$

The coefficient of

is equal to

On the other hand, the left hand side of (1) can be written as

$\displaystyle \sum_{j=1}^\infty \frac{1}{j}(\sigma_1z-\sigma_2z^2+\ldots+(-1)^{n-1} \sigma_n z^n)^j.$

For each

, the coefficient of

$\displaystyle (\sigma_1z-\sigma_2z^2+\ldots+(-1)^{n-1} \sigma_n z^n)^j$

$\displaystyle \sum_{i_1,\ldots,i_n} (-1)^{i_2+i_4+i_6+\ldots} \frac{j!}{i_1!\cdots i_n!}\sigma_1^{i_1} \cdots \sigma_n^{i_n},$

where the summation is extended over all

-tuple $(i_1,\ldots,i_n)$ whose entries are non-negative integers, such that

$\displaystyle i_1+i_2+\ldots+i_n = j$
$\displaystyle i_1+2i_2+\ldots +ni_n = k.$

So the coefficient of

in the left hand side of (1) is

$\displaystyle \sum_{j=1}^\infty \sum_{i_1,\ldots,i_n} (-1)^{i_2+i_4+i_6+\ldots} \frac{(j-1)!}{i_1!\cdots i_n!}\sigma_1^{i_1} \cdots \sigma_n^{i_n},$

$\displaystyle \sum (-1)^{i_2+i_4+i_6+\ldots} \frac{(i_1+\ldots+i_n-1)!}{i_1!\cdots i_n!}\sigma_1^{i_1} \cdots \sigma_n^{i_n}.$

The last summation is over all $(i_1,\ldots, i_n)\in \mathbb{Z}^n$ with non-negative entries such that $i_1+2i_2+\ldots+ni_n=k$ .

"proof of Waring's formula" is owned by kshum.

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11C08 (Number theory :: Polynomials and matrices :: Polynomials)

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