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Fungrim entry: c687d6

ψ(m) ⁣(zn)=ψ(m) ⁣(z)(1)mm!k=1n1(zk)m+1\psi^{(m)}\!\left(z - n\right) = \psi^{(m)}\!\left(z\right) - {\left(-1\right)}^{m} m ! \sum_{k=1}^{n} \frac{1}{{\left(z - k\right)}^{m + 1}}
Assumptions:mZ0  and  zC  and  nZ0  and  zn{0,1,}m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}
\psi^{(m)}\!\left(z - n\right) = \psi^{(m)}\!\left(z\right) - {\left(-1\right)}^{m} m ! \sum_{k=1}^{n} \frac{1}{{\left(z - k\right)}^{m + 1}}

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

2021-03-15 19:12:00.328586 UTC