Binomial Theorem/Hurwitz's Generalisation/Mistake

Source Work

 * Chapter Two: Information Structures
 * $2.3.$ Trees:
 * $2.3.4.$ Basic Mathematical Properties of Trees
 * $2.3.4.4.$ Enumeration of Trees
 * Exercise $30$: Solution
 * Exercise $30$: Solution

Mistake

 * Use this to prove Hurwitz' generalization of the binomial theorem:
 * $\displaystyle \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} y \paren {y + \paren {1 - \epsilon_1} z_1 - \cdots + \paren {1 - \epsilon_n} z_n}^{n - \epsilon_1 - \cdots - \epsilon_n} = \paren {x + y} \paren {x + y + z_1 + \cdots + z_n}^{n - 1}$

Correction
There is an extraneous $y$ here. It should read:


 * $\displaystyle \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y + \paren {1 - \epsilon_1} z_1 - \cdots + \paren {1 - \epsilon_n} z_n}^{n - \epsilon_1 - \cdots - \epsilon_n} = \paren {x + y} \paren {x + y + z_1 + \cdots + z_n}^{n - 1}$