Falling factorial

The falling factorial $x^{\underline{k}}$ for nonnegative integer $k$ is given by $$x^{\underline{k}}=x(x-1)\ldots (x-k+1).$$ If $k$ is not an integer, we use the following formula to interpret $x^{\underline{k}}$: $$x^{\underline{k}} = \dfrac{\Gamma(x+1)}{\Gamma(x-k+1)},$$ where $\Gamma$ denotes the gamma function.