three binomial(n,k) definitions

[Henry Gould]

Agreed absolutely. I do not question the definition

(*)    binomial(n,k) = n(n-1)...(n-k+1) / k!

for all complex  n  and integers  k  *  0.

It is only when you try to allow  k < 0 that we get into
problems. So don't get rattled that the visible world will
go dark. The binomial polynomial will be with us
till infimnity freezes over!

Seriatim,

Henry Gould


Brendan McKay wrote:

> One can have fun discussing how to define binomial(n,k)
> when k is not a non-negative integer.  However, modern
> mathematics will crumble, libraries will self-destruct
> and mothers will stop loving their children if a
> definition like
>
>    binomial(n,k) = if(k<0|k>n,0,n!/(k!*(n-k)!))
>
> is adopted.  The reason is that the BINOMIAL THEOREM
> requires
>    binomial(n,k) = n*(n-1)*...*(n-k+1) / k!
> for all non-negative integers k, REGARDLESS of n and even
> if n is a complex number!  Vast tracts of mathematics,
> including all the analysis that ultimately derives from
> Taylors theorem, books full of combinatorial enumeration,
> etc, etc, etc, ETC, depend on this.
>
> So let's not get too carried away here!
>
> Brendan.