an algebraic identity leading to Wilson’s theorem

For any positive integer $n$ and any real or complex number $x$ ,

\\sum_{k=0}^{n}(-1)^{k}{n\\choose k}(x-k)^{n}=n!

Furthermore, if $n>m$ then

\\sum_{k=0}^{n}(-1)^{k}{n\\choose k}(x-k)^{m}=0

References

1 S. M Ruiz. An algebraic identity leading to Wilson’s theorem. The Mathemtical Gazette, 80(489):579–582, November 1996. http://arxiv.org/abs/math.GM/0406086math.GM/0406086.

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