three binomial(n,k) definitions

[Karol PENSON]

    Dear Michael, I am truly chocked .
    How to remedy this ?
                         Karol




On Fri, 13 Dec 2002, Michael Somos wrote:

> Date: Fri, 13 Dec 2002 14:19:21 -0500
> From: Michael Somos <somos at grail.cba.csuohio.edu>
> To: seqfan at ext.jussieu.fr
> Subject: three binomial(n,k) definitions
> 
> 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"? More to the point, when the
> binomial(n,k) is used in a formula which involves n<0 or
> k<0, how are we to guess which of the three interpretations
> is to apply? This also relates to my earlier attempt to
> raise the issue of two-way infinite and one-way infinite
> sequences. Shalom, Michael
> 

-- 
_________________________________________________________________________
Karol A. PENSON
Universite Paris 6              |  Internet : penson at lptl.jussieu.fr.
Lab. Physique Theorique des     |
Liquides                        | http://www.lptl.jussieu.fr/users/penson
4, place Jussieu, Tour 16, Et. 5|      Tel : (33 1) 44 27 72 33
75252 Paris Cedex 05, France    |      Fax : (33 1) 44 27 51 00