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

Gamma function