Binomial Theorem/Abel's Generalisation/Proof 2

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Theorem

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


Proof

From this formula:

$(1): \quad \displaystyle \sum_{k \mathop \in \Z} \binom n k x \left({x + k z}\right)^{k - 1} y \left({y + \left({n - k}\right) z}\right)^{n - k - 1} = \left({x + y}\right) \left({x + y + n z}\right)^{n - 1}$

The given formula:

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

can then be transformed into $(1)$ by



Sources