Max Alekseyev on Fri, 28 Oct 2022 21:34:52 +0200
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Re: binomial coefficient with negative args
- To: Christian Krause <firstname.lastname@example.org>
- Subject: Re: binomial coefficient with negative args
- From: Max Alekseyev <email@example.com>
- Date: Fri, 28 Oct 2022 15:33:13 -0400
- Cc: firstname.lastname@example.org
- Delivery-date: Fri, 28 Oct 2022 21:34:52 +0200
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The extension surely has its ground, but the problem is that it contradicts the conventional use of binomial coefficients in combinatorics. Having binomial(n,k) = 0 for k < 0 allows one to not care much about range for the lower index and consider sums where some terms are zero just because k goes out of range (which is quite convenient). And as Bill said, enabling the possibility binomial(n,k) to be nonzero for k < 0 may break many existing scripts (mine included).
On Fri, Oct 28, 2022 at 2:34 PM Christian Krause <email@example.com
Some more info on the extension: it uses the gamma function as continuous version of factorial. The rest follows from the definition of the gamma function. Besides Wolfram Alpha, also the Maple Calculator app uses this definition:
On Wed, Oct 26, 2022 at 02:54:47PM -0400, Max Alekseyev wrote:
> I worry that this extension is not consistent with the definition of
> binomial(n,k) as the coefficient of x^k in (1+x)^n.
> According to this definition it should be zero for k < 0.
To give an example where that makes a difference:
%1 = 1
%1 = 0