Falling Factorial

The falling factorial, denoted ( x ) n subscript 𝑥 𝑛 (x)_{n} or x n ¯ superscript 𝑥 ¯ 𝑛 x^{\underline{n}} , is the Polynomial

(x)n=x⁢(x−1)⁢(x−2)⁢⋯⁢(x−n+1)=∏k=0n−1(x−k)subscript𝑥𝑛𝑥𝑥1𝑥2⋯𝑥𝑛1superscriptsubscriptproduct𝑘0𝑛1𝑥𝑘(x)_{n}=x(x-1)(x-2)\cdots(x-n+1)=\prod_{k=0}^{n-1}(x-k)

The falling factorials are a Basis for the Polynomial Ring.