On Sums of Powers of Odd Integers

DOI：10.3770/j.issn:2095-2651.2013.06.003

 作者 单位 郭松柏 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 沈有建 海南师范大学数学与统计学院, 海南 海口 571158

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.

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.