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Fungrim entry: 7b724b

ψ(m) ⁣(nz)=1nm+1k=0n1ψ(m) ⁣(z+kn)\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)
Assumptions:mZ1  and  nZ1  and  zC  and  nz{0,1,}m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}
\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    Formula(Equal(DigammaFunction(Mul(n, z), m), Mul(Div(1, Pow(n, Add(m, 1))), Sum(DigammaFunction(Add(z, Div(k, n)), m), For(k, 0, Sub(n, 1)))))),
    Variables(m, n, z),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)), Element(z, CC), NotElement(Mul(n, 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.

2021-03-15 19:12:00.328586 UTC