Fungrim entry: 7a36e5

G\!\left(z + n\right) = \left[\prod_{k=1}^{n} {\left(z + k - 1\right)}^{n - k}\right] {\left(\Gamma(z)\right)}^{n} G(z)

Assumptions:

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

TeX:

G\!\left(z + n\right) = \left[\prod_{k=1}^{n} {\left(z + k - 1\right)}^{n - k}\right] {\left(\Gamma(z)\right)}^{n} G(z)

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

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7a36e5"),
    Formula(Equal(BarnesG(Add(z, n)), Mul(Mul(Brackets(Product(Pow(Sub(Add(z, k), 1), Sub(n, k)), For(k, 1, n))), Pow(Gamma(z), n)), BarnesG(z)))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)), Element(n, ZZGreaterEqual(0)))))

Topics using this entry

Barnes G-function