Euler beta function

Function

definition $ {\mathrm B}: \{z\ |\ \mathfrak{R}(z) > 0 \}^2 \to \mathbb C$
definition $ {\mathrm B}(p,q) := \int_0^1 \tau^{p-1}(1-\tau)^{q-1}\,\mathrm d\tau $

Theorems

For natural numbers

  • ${\large{n \choose k}}=(n+1)\cdot\dfrac{1}{{\mathrm B}(n-k+1,k+1)}$
  • $\dfrac{1}{{\mathrm B}(x,y)} = \frac{x\,y}{x+y} \prod_{n=1}^\infty \left( 1 + \dfrac{x\,y}{n\,(x+y+n)}\right)$

Reference

Wikipedia: Beta function


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