Fungrim entry: c733f7

\left(z\right)_{k} = \frac{\Gamma\!\left(z + k\right)}{\Gamma\!\left(z\right)}

Assumptions:

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

TeX:

\left(z\right)_{k} = \frac{\Gamma\!\left(z + k\right)}{\Gamma\!\left(z\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`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("c733f7"),
    Formula(Equal(RisingFactorial(z, k), Div(GammaFunction(Add(z, k)), GammaFunction(z)))),
    Variables(z, k),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)), NotElement(Add(z, k), ZZLessEqual(0)))))

Topics using this entry

Factorials and binomial coefficients