an algebraic identity leading to Wilson’s theorem

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

$$\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0pt}{}{n}{k}\right){(x-k)}^n=n!$$

Furthermore, if $n>m$ then

$$\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0pt}{}{n}{k}\right){(x-k)}^m=0$$

Title	an algebraic identity leading to Wilson’s theorem
Canonical name	AnAlgebraicIdentityLeadingToWilsonsTheorem
Date of creation	2013-03-22 14:31:38
Last modified on	2013-03-22 14:31:38
Owner	GeraW (6138)
Last modified by	GeraW (6138)
Numerical id	16
Author	GeraW (6138)
Entry type	Result
Classification	msc 11B65
Classification	msc 05A10
Related topic	Factorial
Related topic	WilsonsTheorem