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Fungrim entry: 554ac2

ψ ⁣(zn)=ψ ⁣(z)k=1n1zk\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}
Assumptions:zC  and  nZ0  and  zn{0,1,}z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}
TeX:
\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Sumnf(n)\sum_{n} f(n) Sum
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("554ac2"),
    Formula(Equal(DigammaFunction(Sub(z, n)), Sub(DigammaFunction(z), Sum(Div(1, Sub(z, k)), For(k, 1, n))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)), NotElement(Sub(z, n), 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.

2020-08-27 09:56:25.682319 UTC