Fungrim entry: 72db94

\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}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\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}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("72db94"),
    Formula(Equal(BetaFunction(n, b), Cases(Tuple(UnsignedInfinity, Element(Neg(b), Range(0, Sub(n, 1)))), Tuple(Div(1, Mul(n, Binomial(Sub(Add(n, b), 1), n))), Otherwise)))),
    Variables(n, b),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(b, CC))))

Topics using this entry

Beta function