Sums of powers

Bernoulli link

So we have a neat generating function. So what? In this case the form links into a well-known generating function for the Bernoulli polynomials (what?): \[\frac{t e^{zt}}{e^t-1} = \sum\limits_{n=0}^\infty B_n(z) \frac{t^n}{n!}\]

Hopefully you can see the relation with our power-sum generating function:\[tG_N(t)=\sum\limits_{k=0}^\infty [B_k(N+1)-B_k(0)] \frac{t^k}{k!}=\sum\limits_{k=0}^\infty\frac{t^{k+1}}{k!}S_N^{(k)}\]

Lining up the powers in \(t\) and equating we get: \[[B_{k+1}(N+1)-B_{k+1}(0)]\frac{1}{(k+1)!}=S_N^{(k)}\frac{1}{k!}\]

Rearranged, this is: \[S_N^{(k)}=\frac{B_{k+1}(N+1)-B_{k+1}(0)}{k+1}\]