Editing Binomial coefficient (section)

=== Two real or complex valued arguments ===
The binomial coefficient is generalized to two real or complex valued arguments using the [[gamma function]] or [[beta function]] via
: <math>{x \choose y}= \frac{\Gamma(x+1)}{\Gamma(y+1) \Gamma(x-y+1)}= \frac{1}{(x+1) \Beta(y+1, x-y+1)}.</math>
This definition inherits these following additional properties from <math>\Gamma</math>:
: <math>{x \choose y}= \frac{\sin (y \pi)}{\sin(x \pi)} {-y-1 \choose -x-1}= \frac{\sin((x-y) \pi)}{\sin (x \pi)} {y-x-1 \choose y};</math>
moreover,
: <math>{x \choose y} \cdot {y \choose x}= \frac{\sin((x-y) \pi)}{(x-y) \pi}.</math>

The resulting function has been little-studied, apparently first being graphed in {{Harv|Fowler|1996}}. Notably, many binomial identities fail: <math display="inline">\binom{n }{ m} = \binom{n }{ n-m}</math> but <math display="inline">\binom{-n}{m} \neq \binom{-n}{-n-m}</math> for ''n'' positive (so <math>-n</math> negative). The behavior is quite complex, and markedly different in various octants (that is, with respect to the ''x'' and ''y'' axes and the line <math>y=x</math>), with the behavior for negative ''x'' having singularities at negative integer values and a checkerboard of positive and negative regions:
* in the octant <math>0 \leq y \leq x</math> it is a smoothly interpolated form of the usual binomial, with a ridge ("Pascal's ridge").
* in the octant <math>0 \leq x \leq y</math> and in the quadrant <math>x \geq 0, y \leq 0</math> the function is close to zero.
* in the quadrant <math>x \leq 0, y \geq 0</math> the function is alternatingly very large positive and negative on the parallelograms with vertices <math display="block">(-n,m+1), (-n,m), (-n-1,m-1), (-n-1,m)</math>
* in the octant <math>0 > x > y</math> the behavior is again alternatingly very large positive and negative, but on a square grid.
* in the octant <math>-1 > y > x + 1</math> it is close to zero, except for near the singularities.