Talk:Binomial Theorem/Hurwitz's Generalisation

The source (Knuth) is incorrect.

Based on Binomial Theorem/Abel's Generalisation, (and Hurwitz himself), that lone y does not belong there.

Abel's Generalisation

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

Hurwitz's Generalisation (incorrect - including the lone $y$)

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

Source 1 - Exercise 30:

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $51$

Hurwitz's Generalisation (corrected - lone y removed)

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

Letting Hurwitz speak for himself (page 202) [1]

"Diese vereinfacht sich noch, wenn man $\nu$ durch $\nu - \paren {x_1 + x_2 + \cdots + x_n}$ ersetzt. Dadurch erhalt man namlich"

$\ds \sum_k \paren {\mu + x_{\alpha 1 } + x_{\alpha 2 } + \cdots + x_{\alpha \lambda} }^{\lambda - 1} \paren {\nu - x_{\alpha 1} - x_{\alpha 2} - \cdots - x_{\alpha \lambda} }^{n - \lambda} = \frac 1 \mu \paren {\mu + \nu}^n$

I saw your humorous comment on Binomial Theorem/Abel's Generalisation/Proof 3 and that's what generated this.

--Robkahn131 (talk) 13:58, 21 June 2020 (EDT)

Good catch. --prime mover (talk) 15:22, 21 June 2020 (EDT)

I've checked http://www.kcats.org/csci/464/doc/knuth/errata/all1.pdf and I've got a feeling that this is a mistake in TAOCP which has not been caught. It does have a report of a mistake on p. 399 but it appears this correction is also incorrect. Do you want to write to DEK and point it out to him, since you found it? You don't get paid for it any more, but it's certainly kudos points. --prime mover (talk) 15:53, 21 June 2020 (EDT)

Will do. --Robkahn131 (talk) 16:16, 21 June 2020 (EDT)