Binomial expansion

Generalized binomial coëfficient

Let 𝑅 be a commutative ring in which ℕ is invertible, 𝛼 ∈𝑅, and 𝑘 ∈ℕ0. Then the generalized binomial coëfficients are defined by num

(𝛼𝑘)=𝛼𝑘――𝑘!=𝛼(𝛼−1)⋯(𝛼−𝑘+1)𝑘(𝑘−1)⋯1

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

(1+𝑋)𝛼=∞∑𝑘=0(𝛼𝑘)𝑋𝑘

Properties

1. If 𝛼 ∈ℤ then (𝛼𝑘) =(−𝛼+𝑘−1𝑘)

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