Beta function

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B(p,\, q) = \frac{\Gamma(p) \, \Gamma(q)}{\Gamma(p+q)}, \,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \Gamma(s) \,} is the Gamma function.

It is related to the binomial coefficients by (as a result of the universally adopted [unfortunate] notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \Gamma(n) \,=\, (n-1)! \,} due to Legendre, instead of Gauss's simpler Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \Gamma(n) \,=\, n! \,} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 \N_+. \,}

See also

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