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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:nZ1andbCn \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
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
UnsignedInfinity~{\tilde \infty} Unsigned infinity
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
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:
Entry(ID("72db94"),
    Formula(Equal(BetaFunction(n, b), Cases(Tuple(UnsignedInfinity, Element(Neg(b), ZZBetween(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

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

2019-08-21 11:44:15.926409 UTC