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Fungrim entry: 7a36e5

G ⁣(z+n)=[k=1n(z+k1)nk](Γ(z))nG(z)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:zC  and  z{0,1,}  and  nZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
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}
Fungrim symbol Notation Short description
BarnesGG(z)G(z) Barnes G-function
Productnf(n)\prod_{n} f(n) Product
Powab{a}^{b} Power
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    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)))))

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2021-03-15 19:12:00.328586 UTC