Fungrim entry: a7dbf6

\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}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\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}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("a7dbf6"),
    Formula(Equal(BetaFunction(Neg(n), b), Cases(Tuple(Div(Pow(-1, b), Mul(b, Binomial(n, b))), Element(b, Range(1, n))), Tuple(UnsignedInfinity, Otherwise)))),
    Variables(n, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(b, CC))))

Topics using this entry

Beta function