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Fungrim entry: 72db94

B ⁣(n,b)={~,b{0,1,,n1}1n(n+b1n),otherwise\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:nZ1  and  bCn \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
\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}
Fungrim symbol Notation Short description
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
UnsignedInfinity~{\tilde \infty} Unsigned infinity
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Binomial(nk){n \choose k} Binomial coefficient
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(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))))

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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