Another Approach on Power Sums

Abstract

We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express the monomials~$n^m$. Once the expansion coefficients are determined, we can express the $m$-th power sums~$S^{(a)}_m(n)$ of any order $a$, \[ S^{(a)}_m(n) = \sum_{\nu_a = 1}^n \cdots \sum_{\nu_2 = 1}^{\nu_3} \sum_{\nu_1=1}^{\nu_2} \nu_1^m, \] in a very convenient way by exploiting the summation property of the $\psi_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk} \psi_k^{(a)}(n). \]