three binomial(n,k) definitions

[Henry Gould]

As for the gamma function, as someone has suggested as
a way to extend the factorials and binomial coefficient.,
this would not suit everyone either since the
classical gamma function is but one of infinitely many possible
generalizations the factorial. People justify it use because of
the Bohr-Mollerup-Artin theorem about convex functions.
But that can be looked at as an esthetic choice.
The definition can vary according to the circumstances.
Maybe it is rather like Bohr's Complementarity Principle:
waves versus photons. Light may be neither, or both or
something transcending our particular definitions.

Regards,

Henry Gould

"Brian L. Galebach" wrote:

> Oops.  Missed a parenthesis and messed up my variables.
>
> binomial(n,k) = lim(a->n,b->k) gamma(a+1)/(gamma(b+1)*gamma(a-b+1)).
>
> -----Original Message-----
> From: Brian L. Galebach [mailto:briang at SEGmail.com]
> Sent: Friday, December 13, 2002 3:08 PM
> To: 'gould at math.wvu.edu'; 'Michael Somos'; 'seqfan at ext.jussieu.fr'
> Cc: 'briang at ProbabilitySports.com'
> Subject: RE: three binomial(n,k) definitions
>
> 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