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On the sum of $k$-th powers in terms of earlier sums
| Content Provider | Semantic Scholar |
|---|---|
| Author | Miller, Steven J. Treviño, Enrique |
| Copyright Year | 2019 |
| 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. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/SumkPowersRevisionArXiv.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/1912.07171v1.pdf |
| Alternate Webpage(s) | http://campus.lakeforest.edu/trevino/SumkPowers.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
