| 郭松柏,沈有建.奇数方幂和的通项公式[J].数学研究及应用,2013,33(6):666~672 |
| 奇数方幂和的通项公式 |
| On Sums of Powers of Odd Integers |
| 投稿时间：2012-09-28 修订日期：2013-04-18 |
| DOI：10.3770/j.issn:2095-2651.2013.06.003 |
| 中文关键词: 奇数 方幂和 二项式定理 叠加法. |
| 英文关键词:odd number sums of powers binomial theorem superposition method. |
| 基金项目:国家自然科学基金(Grant No.111004). |
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| 中文摘要: |
| 本文利用叠加法简洁地证明了$\sum\limits^{n}_{i=1}(2i-1)^{2k-1}$为$n^2$与$n^2$的$k-1$次有理多项式的乘积,$\sum\limits^{n}_{i=1}(2i-1)^{2k}$为$n(2n-1)(2n+1)$与$(2n-1)(2n+1)$的$k-1$次有理多项式的乘积,并给出了相应的有理多项式的系数的递推计算公式. |
| 英文摘要: |
| In this paper, by using superposition method, we aim to show that $\sum_{i = 1}^n {(2i - 1)^{2k -1}} $ is the product of $n^2$ and a rational polynomial in $n^2$ with degree $k - 1$, and that $\sum_{i = 1}^n {(2i - 1)^{2k}}$ is the product of $n(2n - 1)(2n + 1)$ and a rational polynomial in $(2n - 1)(2n + 1)$ with degree $k-1$. Moreover, recurrence formulas to compute the coefficients of the corresponding rational polynomials are also obtained. |
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