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Proof of an expression of which the binomial formula is a special case. (Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist.) (German)
J. Reine Angew. Math. 1, 159-160 (1826).
Abel proves the identity $$\align (x+α)^n & = x^n + \frac{n}1 α(x+β)^{n-1} + \frac{n \cdot (n-1)}{1 \cdot 2} α(α- 2β) (x+2β)^{n-1} + \ldots \ & \quad + \frac{n}1 α(α- (n-1)β)^{n-2} (x + (n-1)β) +α(α- nβ)^{n-1}, \endalign$$ which contains as a special case ($β= 0$) the usual binomial expansion $(x+α)^n$ for positive integers $n$.
Reviewer: Franz Lemmermeyer (Jagstzell) (2014)
Classification: H20 A30
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