(related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)
Let \(x\) be a real (or complex) number and let \(-x\) be its additive inverse (i.e. \(x + (-x)=0\)). Then it follows immediately from the definition of the falling and rising factorial powers that
\[(-x)^{\underline k}=(-x)(-x-1)\cdots(-x-k+1)=(-1)^kx(x+1)\cdot(x+k-1)=(-1)^k x^{\overline k}.\]