This question is so naive that it could have been asked before on this site. If so, I'll delete it.


Among beautiful formula, I like a lot this one:
$$\left(\sum\_{n=1}^Nn\right)^2=\sum\_{n=1}^Nn^3.$$
Is there any other algebraic relation between the polynomials $P\_k$ defined by
$$P\_k(N):=\sum\_{n=1}^Nn^k \qquad?$$
I suspect yes, because $1,P\_0,P\_1,\ldots$ is a basis of ${\mathbb Q}[X]$ (but not a basis of the $\mathbb Z$-module ${\mathbb Z}[X]$), and if one replace $P\_{\ell m}$ by $P\_\ell P\_m$, we get another basis. But are there nice relations?



Relations between sums of powers
<http://en.wikipedia.org/wiki/Faulhaber%27s_formula>


QUOTE:


Faulhaber observed that if $p$ is odd, then


$$1^p + 2^p + 3^p + \cdots + n^p$$


is a polynomial function of


$$a=1+2+3+\cdots+n= \frac{n(n+1)}{2}.$$


END OF QUOTE


QUOTE:


Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Math. Comp. (American Mathematical Society) 61 (203): 277–294. arXiv:math.CA/9207222. doi:10.2307/2152953. JSTOR 2152953. The arxiv.org paper has a misprint in the formula for the sum of 11th powers, which was corrected in the printed version.


END OF QUOTE


CORRECT VERSION:


<http://www-cs-faculty.stanford.edu/~knuth/papers/jfsp.tex.gz>


