Binomial Theorem/Hurwitz's Generalisation/Mistake

Source Work

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.):

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

Mistake

Use this to prove Hurwitz' generalization of the binomial theorem:

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

Correction

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

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

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.)