three binomial(n,k) definitions

[Brian L. Galebach]

Possibly a consistent treatment could be derived through use of the gamma
function.  Then we can let

binomial(n,k) = lim(a->n,b->k) gamma(n+1)/(gamma(k+1)*gamma(n-k+1).

The limits are necessary since the values of gamma for non-negative integers
goes to infinity.

Just an idea.

Brian

-----Original Message-----
From: Henry Gould [mailto:gould at math.wvu.edu]
Sent: Friday, December 13, 2002 2:45 PM
To: Michael Somos; seqfan at ext.jussieu.fr; gould at math.wvu.edu
Subject: Re: three binomial(n,k) definitions


Yes, there has never been agreement aboiuyt what to do with the
binomial coefficient C(n,k) when  k is negative.
Please see my paper
Note on a Paper of Klamkin concerning Stirling numbers, Amer. Math.
Monthly, 68(1961), 477-479.
Reviews in: MR 23(1962), #A1548; Zentralblatt 146(1968), pp.50-51; Ref
Zhur. 1962, #5A163.

I showed there that Klamkin (1957)made a mistake of this sort when  he
used manipulations
with binomial identities and thought to have found a nice way to express
a certain series
using Stirling numbers of the  f i r s t  kind, but when I  examined his
work I found that
nothing new could be gotten and the work was a mere tautology, due to
the ambiguity of
C(-n,-k) when n and k are positive.

Moreover, if you examine John Riordan's  famous book "An Introduction to
Combinatorial
Analysis," published by Wiley, N.Y., 1958, in particular page 5, he
chose one way to give a
meaning for C(-n,-k) and yet it is easily and equally argueable to have
the negative of this.
My conclusion has been that since the only number X having the property
that X = -X is
zero, then a truly suitable and satisfying continuation or
generalization is not really
possible. I have never seen any definition that breaches the gap.

This subject is dear to my heart, and I would enjoy hearing from anyone
about it.

Sincerely,

Henry W. Gould

Michael Somos 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"? 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