Beta function

This article page is a stub, please help by expanding it.

The [reciprocal of] the beta function extends the binomial coefficients to all the complex numbers.

$B(p,\,q)={\frac {\Gamma (p)\,\Gamma (q)}{\Gamma (p+q)}},\,$

where $\scriptstyle \Gamma (s)\,$ is the Gamma function.

It is related to the binomial coefficients by (as a result of the universally adopted [unfortunate] notation $\scriptstyle \Gamma (n)\,=\,(n-1)!\,$ due to Legendre, instead of Gauss's simpler $\scriptstyle \Gamma (n)\,=\,n!\,$)

$B(m,\,n)={\frac {(m-1)!\,(n-1)!}{(m+n-1)!}}={\binom {m+n}{m}}^{-1}={\binom {m+n}{n}}^{-1},\quad m,\,n\in \mathbb {N} _{+}.\,$

See also

• Weisstein, Eric W., Beta Function, from MathWorld—A Wolfram Web Resource.