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Newton-Girard Formulas

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The identities between the symmetric polynomials Pi_k(x_1,...,x_n) and the sums of kth powers of their variables

S_k(x_1,...,x_n)=sum_(j=1)^nx_j^k.
(1)

The identities are given by

(-1)^mmPi_m(x_1,...,x_n)+sum_(k=1)^m(-1)^(k+m)S_k(x_1,...,x_n)Pi_(m-k)(x_1,...,x_n)=0
(2)

for each 1<=m<=n and for an arbitrary number of variables n. The first few identities are

S_1-Pi_1=0
(3)
S_2-S_1Pi_1+2Pi_2=0
(4)
S_3-S_2Pi_1+S_1Pi_2-3Pi_3=0
(5)
S_4-S_3Pi_1+S_2Pi_2-S_1Pi_3+4Pi_4=0.
(6)

See also

Power Sum, Symmetric Polynomial

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References

Séroul, R. "Newton-Girard Formulas." §10.12 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 278-279, 2000.

Referenced on Wolfram|Alpha

Newton-Girard Formulas

Cite this as:

Weisstein, Eric W. "Newton-Girard Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-GirardFormulas.html

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