Falling factorial

Definition

Let $x$ be an element of a commutative ring, for example a real number, and let $k$ be a natural number. The falling factorial of $x$ of order $k$ is $$(x)_k = x (x - 1) (x - 2) \cdots (x - k + 1) = \prod_{i=0}^{k-1} (x - i).$$ For $k = 0$ the product is empty, and $(x)_0 = 1$.