Binomial coefficient over the complex numbers
Function
| definition | $??$ |
| definition | ${x\choose{y}} := \dfrac{1}{\Gamma(y+1)}\dfrac{\Gamma(x+1)}{\Gamma((x+1)-(y+1)+1)}$ |
$(1+z)^s$
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.
(*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
Reference
Wikipedia: Binomial coefficient
Requirements
