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The [reciprocal of] the beta function extends the binomial coefficients to all the complex numbers.
![{\displaystyle B(p,\,q)={\frac {\Gamma (p)\,\Gamma (q)}{\Gamma (p+q)}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e5a0beccc291c9ea005a23621a4144332f661587)
where
is the Gamma function.
It is related to the binomial coefficients by (as a result of the universally adopted [unfortunate] notation
due to Legendre, instead of Gauss's simpler
)
See also