===== Binomial coefficient over the complex numbers =====
==== Function ====
| @#FF9944: definition  | @#FF9944: $??$ |
| @#FF9944: definition  | @#FF9944: ${x\choose{y}} := \dfrac{1}{\Gamma(y+1)}\dfrac{\Gamma(x+1)}{\Gamma((x+1)-(y+1)+1)}$ |

-----
$(1+z)^s$

Sum[Gamma[1 + x]/(Gamma[1 + x - y] Gamma[1 + y])z^y, {y, 0, \[Infinity]}]

=== Discussion ===
For natural numbers $n\ge{k}$ we get

${n\choose{k}} = \dfrac{1}{k!}\dfrac{n!}{(n-k)!}$

=== Theorems ===
${m\choose{m}} = 1$

${m\choose{0}} = 1$

== Pascal's identity (recursive formula) ==
${n\choose{k}} = {n\choose{k-1}} + {n-1\choose{k-1}}$

From this it's also clear that ${n\choose{k}}$ is a sum of 1's, i.e. an integer.

<code>
(*Random generalization of linear such recursive schemes*)

f[m_, 0] = a[m];
f[0, m_] = b[m];
f[m_, m_] = c[m];

f[n_, k_] := A*f[n, k - 1] + B*f[n - 1, k] + C*f[n - 1, k - 1]

Table[{n, k, f[n, k] // sexy}, {n, 1, 4}, {k, 1, 4}] // TableForm
</code>

=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Binomial_coefficient#Two_real_or_complex_valued_arguments|Binomial coefficient]]

-----
=== Requirements ===
[[Gamma function]]