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

Steven J. Miller; Enrique Treviño

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arXiv · 1912.07171

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

Steven J. Miller·Enrique Treviño

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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.

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BibTeX RIS

Steven J. Miller, Enrique Treviño. 2019-12-16. On the sum of $k$-th powers in terms of earlier sums. https://arxiv.org/abs/1912.07171

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