three binomial(n,k) definitions

Michael Somos somos at
Mon Dec 16 18:25:01 CET 2002


     Frank Ruskey wrote :

> I agree with Brendan that we should not play with the
> standard definition of the binomial coefficients.

As far as I know, there is *no* official standard definitions
in mathematics. I would be pleased to be corrected on this.
There are no official standard theorems. There is no official
standard mathematics. What we have now are more or less common
conventions which people agree on most of the time. Just for
example, let us take :

> I.e., binomial(z,k) is 0 if k < 0
> and the degree k polynomial z(z-1)...(z-k+1)/k! otherwise.

I think that a case could be made that binomial(z,k) should be
*undefined* for k not a nonnegative integer. However, that is
a mathematical definition. What a particular implementation
does in case k is not a nonnegative integer is another matter.
It could generate some kind of error, or else return 0 or even
have stranger behavior. A mathematical definition is a guide,
but does not completely determine what any implmentation does.

I think that there is a need for a function conventionally
denoted C(n,k) defined by n!/(k!(n-k)!) only if 0<=k<=n are
integers and zero otherwise. The mathematical function could
be undefined outside the range 0<=k<=n, but since C(n,k) is
never 0 where it is conventionally defined, then C(n,k)=0 can
be used to indicate that n and k are outside that range.

As far as I know, this function is not commonly available in
CAS, unlike binomial(n,k).

> Also, are there instances in OEIS where non-standard
> definitions of binomial(n,k) are used?  Such usage should
> definitely be noted.

I agree completely here. I suggest that C(n,k) should be used
for if(k<0|k>n,0,n!/(k!*(n-k)!)) and binomial(n,k) should be
used for if(k<0,0,prod(i=0,k-1,n-i)/k!). These definitions
agree for n and k nonnegative but binomial(n,k) extends the
definition to k nonegative. If the Maple or any other definition
is used, then it should be so noted. Shalom, Michael

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