On the sum of $k$-th powers in terms of earlier sums

TL;DR

This paper extends Faulhaber's conjecture, proving that for any positive integer k, the sum of k-th powers can be expressed as a polynomial in the sums of lower powers, with a more efficient recursive evaluation method.

Contribution

It generalizes Faulhaber's result by showing all power sums can be written as polynomials in the first two power sums, and introduces a recursive formula for efficient computation.

Findings

Sum of k-th powers expressed as polynomial in S_1 and S_2

Recursive formula reduces polynomial complexity

Extension of Faulhaber's conjecture to all k

Abstract

For $k$ a positive integer let $S_k(n) = 1^k + 2^k + \cdots + n^k$, i.e., $S_k(n)$ is the sum of the first $k$-th powers. Faulhaber conjectured (later proved by Jacobi) that for $k$ odd, $S_k(n)$ could be written as a polynomial of $S_1(n)$; for example $S_3(n) = S_1(n)^2$. We extend this result and prove that for any $k$ there is a polynomial $g_k(x,y)$ such that $S_k(n) = g(S_1(n), S_2(n))$. The proof yields a recursive formula to evaluate $S_k(n)$ as a polynomial of $n$ that has roughly half the number of terms as the classical one.