Binomial Theorem/Hurwitz's Generalisation

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Theorem

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

where the summation ranges over all $2^n$ choices of $\epsilon_1, \ldots, \epsilon_n = 0$ or $1$ independently.


Proof

Follows from this formula:

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



Source of Name

This entry was named for Adolf Hurwitz.


Sources

  • 1902: A. HurwitzÜber Abel's Verallgemeinerung der binomischen Formel (Acta Math. Vol. 26: 199 – 203)