Generalized binomial coëfficient

Let $𝑅$ be a commutative ring in which $ℕ$ is invertible, $𝛼 \in 𝑅$ , and $𝑘 \in ℕ 0$ . Then the generalized binomial coëfficients are defined by #m/def/num

(𝛼 𝑘) = 𝛼 𝑘 ―― 𝑘! = 𝛼 (𝛼 - 1) \dots (𝛼 - 𝑘 + 1) 𝑘 (𝑘 - 1) \dots 1

where we have used the notation of a Falling factorial. We then have the generalized binomial expansion

(1 + 𝑋) 𝛼 = \infty \sum 𝑘 = 0 (𝛼 𝑘) 𝑋 𝑘

Properties

If $𝛼 \in ℤ$ then $(𝛼 𝑘) = (- 𝛼 + 𝑘 - 1 𝑘)$

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