Sum of a to the n formulas

This page gathers some formulas related to sums of squares, cubes, and other symmetric polynomials.

Let

$$ \begin{align} P &= a+b+c \\ Q &= ab+bc+ca \\ R &= abc \\ \end{align} $$ Then $$ a^2+b^2+c^2 = P^2-2Q $$ Then $$ \begin{align} PQ &= (a+b+c)(ab+bc+ca) \\ &= a^2(b+c)+b^2(a+c)+c^2(a+b)+3R \\ \end{align} $$ and $$ P^3=a^3+b^3+c^3+3(a^2(b+c)+b^2(a+c)+c^2(a+b))+6R $$ allows us to derive $$ a^3+b^3+c^3 = P^3-3PQ+3R $$ Continuing we get $$ a^4+b^4+c^4 = P^4-4P^2Q+4PR+2Q^2 $$

Here is a numeric demonstration:

package main

import "fmt"

func check(a, b, c float64) {
        p := a + b + c
        q := a*b + b*c + a*c
        r := a * b * c
        sum2 := a*a + b*b + c*c
        sum3 := a*a*a + b*b*b + c*c*c
        sum4 := a*a*a*a + b*b*b*b + c*c*c*c
        f2 := p*p - 2*q
        f3 := p*p*p - 3*p*q + 3*r
        f4 := p*p*p*p - 4*p*p*q + 4*p*r + 2*q*q
        if f2 != sum2 {
                fmt.Printf("f2/sum2 %.2f != %.2f\n", f2, sum2)
        }
        if f3 != sum3 {
                fmt.Printf("f3/sum3 %.2f != %.2f\n", f3, sum3)
        }
        if f4 != sum4 {
                fmt.Printf("f4/sum4 %.2f != %.2f\n", f4, sum4)
        }
}

func main() {
        check(1, 1, 1)
        check(1, 2, 3)
        check(3, 2, 1)
}

(download)

References

Inequalities by Hardy and Polya