On Sums of Powers of Odd Integers

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.