Binomial Theorem/Abel's Generalisation

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Theorem

$\displaystyle \left({x + y}\right)^n = \sum_k \binom n k x \left({x - k z}\right)^{k - 1} \left({y + k z}\right)^{n - k}$

for $n \in \Z_{\ge 0}$ and $x \in \R_{\ne 0}$.


Proof


Historical Note

Niels Henrik Abel presented this generalisation of the Binomial Theorem in $1826$ in the first volume of August Leopold Crelle's Journal für die reine und angewandte Mathematik.


Sources