three binomial(n,k) definitions

[Franklin T. Adams-Watters]

Just to be explicit:

There are two critical properties of the binomial function:

(1) For k >= 0, C(n,k) is a kth degree polynomial (which I won't repeat here).
(2) C(n,k) = C(n-1,k-1) + C(n-1,k).

These uniquely determine the values for n and k negative.  The result is the table returned by Mathematica.

Maple uses the identity 
(3) C(n,k) = C(n,n-k)
instead of (2).  However, over all, (3) is not as important as (2).  In addition, with the Maple definition, C(n,k) is not a polynomial for k < 0.  These two considerations are enough to say that Mathematica is right, and Maple is wrong.

This is not to imply that the function defined by Maple, or any of various other extensions, are of no interest.

Michael Somos <somos at grail.cba.csuohio.edu> wrote:

>seqfan,
>      I was afraid it would come to this. I just came
>to realize that the three natural variations of the
>binomial table are actually in use without any real
>acknowledgment. Here is the three binomial tables :
>
>      binomial(n,k) according to Maple
>
>n\k  -4   -3   -2   -1    0    1    2    3
>   +--------------------------------------
>-4 |  1    0    0    0    1   -4   10  -20
>-3 | -3    1    0    0    1   -3    6  -10
>-2 |  3   -2    1    0    1   -2    3   -4
>-1 | -1    1   -1    1    1   -1    1   -1
> 0 |  0    0    0    0    1    0    0    0
> 1 |  0    0    0    0    1    1    0    0
> 2 |  0    0    0    0    1    2    1    0
> 3 |  0    0    0    0    1    3    3    1
> 
>binomial(n,k) according to PARI and Mathematica
> 
>n\k  -4   -3   -2   -1    0    1    2    3
>   +--------------------------------------
>-4 |  0    0    0    0    1   -4   10  -20
>-3 |  0    0    0    0    1   -3    6  -10
>-2 |  0    0    0    0    1   -2    3   -4
>-1 |  0    0    0    0    1   -1    1   -1
> 0 |  0    0    0    0    1    0    0    0
> 1 |  0    0    0    0    1    1    0    0
> 2 |  0    0    0    0    1    2    1    0
> 3 |  0    0    0    0    1    3    3    1
> 
>binomial(n,k) = if(k<0|k>n,0,n!/(k!*(n-k)!))
>
>n\k  -4   -3   -2   -1    0    1    2    3
>   +--------------------------------------
>-4 |  0    0    0    0    0    0    0    0
>-3 |  0    0    0    0    0    0    0    0
>-2 |  0    0    0    0    0    0    0    0
>-1 |  0    0    0    0    0    0    0    0
> 0 |  0    0    0    0    1    0    0    0
> 1 |  0    0    0    0    1    1    0    0
> 2 |  0    0    0    0    1    2    1    0
> 3 |  0    0    0    0    1    3    3    1
>
>Notice that for n and k nonnegative, they all agree as
>expected. They first two agree when n<0 and k>=0. Now I
>think it is obvious that these are all natural and valid
>sequence tables. Also obvious that they do not agree for
>all integer n and k. Now which should deserve the name
>of "binomial coefficients"?

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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