Fungrim entry: ea26d4

G\!\left(n z\right) = {e}^{\left(\log(A) - 1 / 12\right) \left({n}^{2} - 1\right)} {n}^{{n}^{2} {z}^{2} / 2 - n z + 5 / 12} {\left(2 \pi\right)}^{\left(n - 1\right) \left(1 - n z\right) / 2} \prod_{i=0}^{n - 1} \prod_{j=0}^{n - 1} G\!\left(z + \frac{i + j}{n}\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

References:

https://arxiv.org/abs/math/0308086

TeX:

G\!\left(n z\right) = {e}^{\left(\log(A) - 1 / 12\right) \left({n}^{2} - 1\right)} {n}^{{n}^{2} {z}^{2} / 2 - n z + 5 / 12} {\left(2 \pi\right)}^{\left(n - 1\right) \left(1 - n z\right) / 2} \prod_{i=0}^{n - 1} \prod_{j=0}^{n - 1} G\!\left(z + \frac{i + j}{n}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BarnesG`	$G(z)$	Barnes G-function
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Product`	$\prod_{n} f(n)$	Product
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ea26d4"),
    Formula(Equal(BarnesG(Mul(n, z)), Mul(Mul(Mul(Pow(ConstE, Mul(Sub(Log(ConstGlaisher), Div(1, 12)), Sub(Pow(n, 2), 1))), Pow(n, Add(Sub(Div(Mul(Pow(n, 2), Pow(z, 2)), 2), Mul(n, z)), Div(5, 12)))), Pow(Mul(2, Pi), Div(Mul(Sub(n, 1), Sub(1, Mul(n, z))), 2))), Product(Product(BarnesG(Add(z, Div(Add(i, j), n))), For(j, 0, Sub(n, 1))), For(i, 0, Sub(n, 1)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(1)))),
    References("https://arxiv.org/abs/math/0308086"))

Topics using this entry

Barnes G-function