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Fungrim entry: b4825b

ψ ⁣(n+z)=1z+ψ ⁣(n+1)+k=1((1)k+1ζ ⁣(k+1)+j=1n1jk+1)zk\psi\!\left(-n + z\right) = -\frac{1}{z} + \psi\!\left(n + 1\right) + \sum_{k=1}^{\infty} \left({\left(-1\right)}^{k + 1} \zeta\!\left(k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + 1}}\right) {z}^{k}
Assumptions:nZ0  and  zC  and  z<1n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1
\psi\!\left(-n + z\right) = -\frac{1}{z} + \psi\!\left(n + 1\right) + \sum_{k=1}^{\infty} \left({\left(-1\right)}^{k + 1} \zeta\!\left(k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + 1}}\right) {z}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
    Formula(Equal(DigammaFunction(Add(Neg(n), z)), Add(Add(Neg(Div(1, z)), DigammaFunction(Add(n, 1))), Sum(Mul(Add(Mul(Pow(-1, Add(k, 1)), RiemannZeta(Add(k, 1))), Sum(Div(1, Pow(j, Add(k, 1))), For(j, 1, n))), Pow(z, k)), For(k, 1, Infinity))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC), Less(Abs(z), 1))))

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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