Binomial Theorem/Abel's Generalisation/Proof 3

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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 Hurwitz's Generalisation of Binomial Theorem:

$(1): \quad \left({x + y}\right)^n = \displaystyle \sum x \left({x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}\right)^{\epsilon_1 + \cdots + \epsilon_n - 1} y \left({y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}\right)^{n - \epsilon_1 - \cdots - \epsilon_n}$

Setting $z = z_1 = z_2 = \cdots z_n$ we have:

\(\displaystyle \) \(\) \(\displaystyle \sum x \left({x + \epsilon_1 z + \cdots + \epsilon_n z}\right)^{\epsilon_1 + \cdots + \epsilon_n - 1} y \left({y - \epsilon_1 z - \cdots - \epsilon_n z}\right)^{n - \epsilon_1 - \cdots - \epsilon_n}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum \binom n k x \left({x + k z}\right)^{k - 1} y \left({y - k z}\right)^{n - k}\) where $\epsilon_1 + \cdots + \epsilon_n = k$
\(\displaystyle \) \(=\) \(\displaystyle \sum \binom n {n - k} x \left({x + k z}\right)^{k - 1} y \left({y - k z}\right)^{n - k}\) Symmetry Rule for Binomial Coefficients
\(\displaystyle \) \(=\) \(\displaystyle \sum \binom n k x \left({x - k z}\right)^{k - 1} y \left({y + k z}\right)^{n - k}\)



Hence the result.


Sources