Fungrim entry: e87c43

{z \choose k} = \frac{\Gamma\!\left(z + 1\right)}{\Gamma\!\left(k + 1\right) \Gamma\!\left(z - k + 1\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z - k \notin \{-1, -2, \ldots\}

TeX:

{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\}

Definitions:

Fungrim symbol	Notation	Short description
`Binomial`	${n \choose k}$	Binomial coefficient
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e87c43"),
    Formula(Equal(Binomial(z, k), Div(GammaFunction(Add(z, 1)), Mul(GammaFunction(Add(k, 1)), GammaFunction(Add(Sub(z, k), 1)))))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)), NotElement(Sub(z, k), ZZLessEqual(-1)))))

Topics using this entry

Factorials and binomial coefficients