Is there a general product formula for $\sum\limits_{k=1}^{n} k^p$

Question

I'm familiar with Faulhaber's formula to express this sum as a much simpler one, but it appears that for any $p$ there's a product formula in $n$ for the sum e.g.:

$$\begin{align} & \sum\limits_{k=1}^{n} k^1=\frac{n(n+1)}{2} \\ & \sum\limits_{k=1}^{n} k^2=\frac{n(n+1)(2n+1)}{6} \\ & \sum\limits_{k=1}^{n} k^3=\frac{n^2(n+1)^2}{4} \end{align}$$

...and so forth. Is there a general product formula in $p$ and $n$ for this sum?

But in general I would not call it a "product formula". The polynomials usually do not have only linear factors. — GEdgar, Feb 01 '14 at 22:54
See http://math.stackexchange.com/questions/63986/asymptotic-behaviour-of-sums-of-consecutive-powers — Gamma Function, Oct 02 '14 at 20:30

score 4 · Answer 1 · answered Oct 02 '14 at 19:54

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This question was answered in a comment:

Yes: http://en.wikipedia.org/wiki/Faulhaber%27s_formula. – Martín-Blas Pérez Pinilla Feb 1 at 22:31

answered Oct 02 '14 at 19:54

Jessica B

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Is there a general product formula for $\sum\limits_{k=1}^{n} k^p$

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