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Fungrim entry: a7dbf6

B ⁣(n,b)={(1)bb(nb),b{1,2,,n}~,otherwise\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:nZ0  and  bCn \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
\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}
Fungrim symbol Notation Short description
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
UnsignedInfinity~{\tilde \infty} Unsigned infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))

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

2021-03-15 19:12:00.328586 UTC