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Fungrim entry: 90a1e1

k=0m1Γ ⁣(z+km)=(2π)(m1)/2m1/2mzΓ ⁣(mz)\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:zCandmZ1andmz{0,1,}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
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("90a1e1"),
    Formula(Equal(Product(GammaFunction(Add(z, Div(k, m))), Tuple(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)))), GammaFunction(Mul(m, z))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Element(m, ZZGreaterEqual(1)), NotElement(Mul(m, z), ZZLessEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC