Re: binomial coefficient with negative args

[Max Alekseyev <maxale@gmail.com>]

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).

Regards,

Max

On Fri, Oct 28, 2022 at 2:34 PM Christian Krause <me@ckrause.org> wrote:

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:

Wolfram Alpha:

Maple Calculator:

Cheers,

Christian

On Fri, 28 Oct 2022 at 12:38, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:

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:

? sum(i=-5,5,binomial(-1,i)*x^i)==(1+x)^-1+O(x^6)

pari 2.15:
%1 = 1
pari 2.16:
%1 = 0

Cheers,
Bill