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- W4287993758 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." @default.
- W4287993758 created "2022-07-26" @default.
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- W4287993758 date "2019-12-15" @default.
- W4287993758 modified "2023-09-27" @default.
- W4287993758 title "On the sum of $k$-th powers in terms of earlier sums" @default.
- W4287993758 doi "https://doi.org/10.48550/arxiv.1912.07171" @default.
- W4287993758 hasPublicationYear "2019" @default.
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