| 奇数方幂和的通项公式 |
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| 引用本文： | 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. |
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| 作者姓名： | 郭松柏 沈有建 |
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| 作者单位： | 海南师范大学数学与统计学院, 海南 海口 571158; 北京科技大学数学与物理学院, 北京 100083;海南师范大学数学与统计学院, 海南 海口 571158 |
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| 基金项目： | 国家自然科学基金(Grant No.111004). |
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| 摘 要： | In this paper, by using superposition method, we aim to show that ∑^n i=1 （2/- 1）^2k-1 is the product of n2 and a rational polynomial in n2 with degree k- 1, and that ∑^ni=1 （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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| 关 键 词： | 有理多项式 幂和 奇数 递推公式 叠加法 NI N2 n2 |
| 收稿时间： | 2012/9/28 0:00:00 |
| 修稿时间： | 2013/4/18 0:00:00 |
On Sums of Powers of Odd Integers |
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| Songbai GUO and Youjian SHEN.On Sums of Powers of Odd Integers[J].Journal of Mathematical Research with Applications,2013,33(6):666-672. |
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| Authors: | Songbai GUO and Youjian SHEN |
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| Institution: | School of Mathematics and Statistics, Hainan Normal University, Hainan 571158, P. R. China; School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China;School of Mathematics and Statistics, Hainan Normal University, Hainan 571158, P. R. China |
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| Abstract: | 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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| Keywords: | odd number sums of powers binomial theorem superposition method |
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