# Fungrim entry: 90a1e1

$\prod_{k=0}^{m - 1} \Gamma\!\left(z + \frac{k}{m}\right) = {\left(2 \pi\right)}^{\left( m - 1 \right) / 2} {m}^{1 / 2 - m z} \Gamma\!\left(m z\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m z \notin \{0, -1, \ldots\}$
TeX:
\prod_{k=0}^{m - 1} \Gamma\!\left(z + \frac{k}{m}\right) = {\left(2 \pi\right)}^{\left( m - 1 \right) / 2} {m}^{1 / 2 - m z} \Gamma\!\left(m z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m z \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
Product$\prod_{n} f(n)$ Product
Gamma$\Gamma(z)$ Gamma function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
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("90a1e1"),
Formula(Equal(Product(Gamma(Add(z, Div(k, m))), For(k, 0, Sub(m, 1))), Mul(Mul(Pow(Mul(2, Pi), Div(Sub(m, 1), 2)), Pow(m, Sub(Div(1, 2), Mul(m, z)))), Gamma(Mul(m, z))))),
Variables(z, m),
Assumptions(And(Element(z, CC), Element(m, ZZGreaterEqual(1)), NotElement(Mul(m, z), ZZLessEqual(0)))))

## Topics using this entry

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