Big mistake. Never mind. Red-facedly, Dan On 2013-06-26, at 12:38 PM, Dan Asimov wrote:
As I see it, there's only one natural way to extend the binomial symbol n_C_k to almost all reals (and also complexes), and that is via the gamma function 𐅃(z), since for positive integers z we have from 𐅃(z) = (z-1)! that
(*) n_C_k = 𐅃(n+1) / (𐅃(k+1) 𐅃(n-k+1))
This definition would imply that 0_C_0 is equal to 1, and also that for integer n and k then for either
a) n >= 0 AND (k <= -1 OR k >= n+1)
we have n_C_k = 0;
OR
b) n <= -1 AND (k <= -1 AND k >= n+1),
--or more simply--
n+1 <= k <= -1
we have n_C_k = 0.
(Since, the two simple poles in the denominator force the quotient to equal 0, despite the simple pole in the numerator.)
Since the gamma function never takes the value 0 and has poles only at the nonpositive integers, a) and b) should cover all cases where the definition (*) of n_C_k = 0.