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Fungrim entry: 039051

ψ(m) ⁣(z+1)=ψ(m) ⁣(z)+(1)mm!zm+1\psi^{(m)}\!\left(z + 1\right) = \psi^{(m)}\!\left(z\right) + \frac{{\left(-1\right)}^{m} m !}{{z}^{m + 1}}
Assumptions:mZ0  and  zC  and  z{0,1,}m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
\psi^{(m)}\!\left(z + 1\right) = \psi^{(m)}\!\left(z\right) + \frac{{\left(-1\right)}^{m} m !}{{z}^{m + 1}}

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Powab{a}^{b} Power
Factorialn!n ! Factorial
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(Add(z, 1), m), Add(DigammaFunction(z, m), Div(Mul(Pow(-1, m), Factorial(m)), Pow(z, Add(m, 1)))))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(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