# 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.
Title an algebraic identity leading to Wilson’s theorem AnAlgebraicIdentityLeadingToWilsonsTheorem 2013-03-22 14:31:38 2013-03-22 14:31:38 GeraW (6138) GeraW (6138) 16 GeraW (6138) Result msc 11B65 msc 05A10 Factorial WilsonsTheorem