Another Approach on Power Sums

We show that explicit forms for certain polynomials~${\psi }_m^{(a)}(n)$ with the property $${\psi }_m^{(a+1)}(n)=\sum _{\nu =1}^n{\psi }_m^{(a)}(\nu )$$ 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~$S_m^{(a)}(n)$ of any order $a$, $$S_m^{(a)}(n)=\sum _{{\nu }_a=1}^n\cdots \sum _{{\nu }_2=1}^{{\nu }_3}\sum _{{\nu }_1=1}^{{\nu }_2}{\nu }_1^m,$$ in a very convenient way by exploiting the summation property of the ${\psi }_m^{(a)}$, $$S_m^{(a)}(n)=\sum _k{c}_{mk}{\psi }_k^{(a)}(n)\mathrm{.}$$