# Newton-Girard formula for symmetric polynomials

Let ${E}_{k}$ be the elementary symmetric polynomials in $n$ variables and ${S}_{k}$ be defined by

$${S}_{k}({x}_{1},\mathrm{\dots},{x}_{n})=\sum _{i=1}^{n}{x}_{i}^{k}.$$ |

Then the ${S}_{k}$ and ${E}_{k}$ are related as follows:

${S}_{1}$ | $={E}_{1}$ | ||

${S}_{2}$ | $={S}_{1}{E}_{1}-2{E}_{2}$ | ||

${S}_{3}$ | $={S}_{2}{E}_{1}-{S}_{1}{E}_{2}+3{E}_{3}$ | ||

$\mathrm{\vdots}$ | |||

${S}_{k}$ | $=-\left({\displaystyle \sum _{j=1}^{k-1}}{(-1)}^{j}{S}_{k-j}{E}_{j}\right)-{(-1)}^{k}k{E}_{k}$ |

By applying these formulas recursively, ${S}_{k}$ can be expressed solely in terms of the ${E}_{k}$, which is often desirable. For example, since ${S}_{1}={E}_{1}$, ${S}_{2}={E}_{1}^{2}-2{E}_{2}$, and then ${S}_{3}=({E}_{1}^{2}-2{E}_{2}){E}_{1}-{E}_{1}{E}_{2}+3{E}_{3}={E}_{1}^{3}-3{E}_{1}{E}_{2}+3{E}_{3}$, and so on.

Note that ${E}_{0}=1$ and ${E}_{k}=0$ for $k>n$.

Title | Newton-Girard formula for symmetric polynomials^{} |
---|---|

Canonical name | NewtonGirardFormulaForSymmetricPolynomials |

Date of creation | 2013-03-22 15:32:40 |

Last modified on | 2013-03-22 15:32:40 |

Owner | kschalm (9486) |

Last modified by | kschalm (9486) |

Numerical id | 5 |

Author | kschalm (9486) |

Entry type | Theorem |

Classification | msc 11C08 |

Related topic | WaringsFormula |

Related topic | ElementarySymmetricPolynomialInTermsOfPowerSums |