Newton' Sum

by luimichael, Mar 5, 2017, 3:35 AM

Just learned the Newton's Sum which is useful in solving higher order system of equations.

Let $a=x+y+z$, $b=xy+yz+zx$ and $c=xyz$.

Then x, y, z are the roots of $u^3-au^2+bu-c=0$.

Let $T_n = x^n+y^n+z^n$ for $n \ge 0$.

$u^{n+3}=au^{n+2}-bu^{n+1}+cu^n$ ---(*) for $n \ge 0$

Put u = x, y, z into (*) and sum up we get $T_{n+3}=aT_{n+2}-bT_{n+1}+cT_{n}$.

Note that $T_0 = 3$, $T_1=a$ and $T_2=(x+y+z)^2-2(xy+yz+zx)=a^2-ab$.

$T_3 = aT_2-bT_1+cT_0 =a(a^2-2b)-b(a)+c(3)=a^3-3ab+3c$

$T_4=aT_3-bT_2+cT_1=a(a^3-3ab+3c)-b(a^2-2b)+ac=a^4-4a^2b+4ac+2b^2$

$T_5=aT_4-bT_3+cT_2=a^5-5a^3b+5a^2c+5ab^2-5bc$

$T_6=aT_5-bT_4+cT_3=a^6-6a^4b+6a^3c+9a^2b^2-12abc-2b^3+3c^2$

$T_7=aT_6-bT_5+cT_4=a^7-7a^5b+7a^4c+14a^3b^2-21a^2bc-7ab^3+7ac^2+7b^2c$.

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