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.

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

where $\mathrm{\Gamma }(s)$ is the Gamma function.

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

$B(m,n)=\frac{(m-1)!(n-1)!}{(m+n-1)!}={(\genfrac{}{}{0ex}{}{m+n}{m})}^{-1}={(\genfrac{}{}{0ex}{}{m+n}{n})}^{-1},m,n\in {\mathbb{\mathbb{N}}}_{+}.$

See also

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