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Fungrim entry: 22ee07

(zk)=(zk+1)kk!{z \choose k} = \frac{\left(z - k + 1\right)_{k}}{k !}
Assumptions:zC  and  kZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}
{z \choose k} = \frac{\left(z - k + 1\right)_{k}}{k !}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Factorialn!n ! Factorial
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Binomial(z, k), Div(RisingFactorial(Add(Sub(z, k), 1), k), Factorial(k)))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)))))

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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