# Beta function

## Definitions

Symbol: BetaFunction $\mathrm{B}\!\left(a, b\right)$ Beta function
Symbol: IncompleteBeta $\mathrm{B}_{x}\!\left(a, b\right)$ Incomplete beta function
Symbol: IncompleteBetaRegularized $I_{x}\!\left(a, b\right)$ Regularized incomplete beta function

## Main formulas

$\mathrm{B}\!\left(a, b\right) = \frac{\Gamma(a) \Gamma(b)}{\Gamma\!\left(a + b\right)}$
$I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}$
$\mathrm{B}_{0}\!\left(a, b\right) = 0$
$\mathrm{B}_{1}\!\left(a, b\right) = \mathrm{B}\!\left(a, b\right)$
$I_{0}\!\left(a, b\right) = 0$
$I_{1}\!\left(a, b\right) = 1$

## Integral representations

$\mathrm{B}\!\left(a, b\right) = \int_{0}^{1} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt$
$\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt$
$\mathrm{B}_{x}\!\left(a, b\right) = \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt$
$I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt$

## Hypergeometric representations

$\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)$

## Symmetry

$\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(b, a\right)$
$I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)$
$\mathrm{B}\!\left(a, b\right) \mathrm{B}\!\left(a + b, c\right) = \mathrm{B}\!\left(b, c\right) \mathrm{B}\!\left(a, b + c\right)$

## Integer parameters

$\mathrm{B}\!\left(m, n\right) = \frac{\left(m - 1\right)! \left(n - 1\right)!}{\left(m + n - 1\right)!}$
$\mathrm{B}\!\left(m, n\right) = \frac{1}{m {m + n - 1 \choose m}}$
$\mathrm{B}\!\left(n, b\right) = \begin{cases} {\tilde \infty}, & -b \in \{0, 1, \ldots, n - 1\}\\\frac{1}{n {n + b - 1 \choose n}}, & \text{otherwise}\\ \end{cases}$
$\mathrm{B}\!\left(-n, b\right) = \begin{cases} \frac{{\left(-1\right)}^{b}}{b {n \choose b}}, & b \in \{1, 2, \ldots, n\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}$
$\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a$

## Recurrence relations

$\left(a + b\right) \mathrm{B}\!\left(a + 1, b\right) = a \mathrm{B}\!\left(a, b\right)$
$\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(a + 1, b\right) + \mathrm{B}\!\left(a, b + 1\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC