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Fungrim entry: e87c43

(zk)=Γ ⁣(z+1)Γ ⁣(k+1)Γ ⁣(zk+1){z \choose k} = \frac{\Gamma\!\left(z + 1\right)}{\Gamma\!\left(k + 1\right) \Gamma\!\left(z - k + 1\right)}
Assumptions:zC  and  kZ0  and  zk{1,2,}z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - k \notin \{-1, -2, \ldots\}
{z \choose k} = \frac{\Gamma\!\left(z + 1\right)}{\Gamma\!\left(k + 1\right) \Gamma\!\left(z - k + 1\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - k \notin \{-1, -2, \ldots\}
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(Binomial(z, k), Div(Gamma(Add(z, 1)), Mul(Gamma(Add(k, 1)), Gamma(Add(Sub(z, k), 1)))))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)), NotElement(Sub(z, k), ZZLessEqual(-1)))))

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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