  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 $\psi_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk}
\psi_k^{(a)}(n). \]
