three binomial(n,k) definitions

James Propp propp at
Mon Dec 16 19:57:37 CET 2002

I've just joined seqfan, so I'm not sure whether the following contribution
to the ongoing discussion will tell people anything new.  But I thought I'd
share it.

(This is a paraphrase of an exchange I had with Ira Gessel a year or so ago.)

For compactness, let B(m,n) be the binomial coefficient {m choose n},
where n is a non-negative integer and m is an arbitrary integer.  
(We define this in the usual way so as to make B(m,n) a polynomial 
in m for each fixed n.)  Since the identities
(1)	B(m,n)=(-1)^n B(-m+n-1,n) 
(2)	B(m,n)=B(m,m-n) 
hold for all m,n that make the expressions well-defined, we might hope
to extend B(m,n) to all of Z x Z so as to prserve this property.  But
this cannot be done: for, applying these two transformations alternately 
three times we get B(m,n) = -B(m,n):

B(m, n) = (-1)^n B(-m+n-1, n) = (-1)^n B(-m+n-1, -m-1)

     = (-1)^(m+n+1) B(-n-1, -m-1)= (-1)^(m+n+1) B(-n-1, -n+m)

     = -B(m, -n+m) = -B(m, n)

So my belief is that the "natural" extension of the binomial coefficient
function is well-defined only up to sign.  If you want to be fancy, you
could say that the natural domain of the binomial coefficient function 
is actually a double cover of Z x Z (though I have no idea what this
might mean, combinatorially).

Jim Propp

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