Newton-Girard Formulas

Author

Eric W. Weisstein
November 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Newton-GirardFormulas.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Newton-GirardFormulas.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Indentities

NewtonGirardIdentities[n_]:=Table[(-1)^m m Π[m]+Total[Table[(-1)^(m+k) S[k]Π[m-k],{k,m}]],{m,n}]/.{S[0]:>0,Π[0]:>1}

NewtonGirardIdentities[7]/.f_[n_]:>Subscript[f,n]//ColumnForm//TraditionalForm

Check

SumOfPowers[n_,x_]:=Total[x^n]

NewtonGirard[n_]:=NewtonGirardIdentities[n]/.{
S[k_]:>SumOfPowers[k,Array[x,n]],
Π[k_]:>SymmetricPolynomial[k,Array[x,n]]
}

NewtonGirard[5]//Simplify//ColumnForm