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Fungrim entry: 739819

k=1nψ ⁣(kn)cos ⁣(2πrkn)=nlog ⁣(2sin ⁣(πrn))\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)
Assumptions:nZ2  and  r{1,2,,n1}n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; r \in \{1, 2, \ldots, n - 1\}
TeX:
\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; r \in \{1, 2, \ldots, n - 1\}
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
Loglog(z)\log(z) Natural logarithm
Sinsin(z)\sin(z) Sine
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
Entry(ID("739819"),
    Formula(Equal(Sum(Mul(DigammaFunction(Div(k, n)), Cos(Div(Mul(Mul(Mul(2, Pi), r), k), n))), For(k, 1, n)), Mul(n, Log(Mul(2, Sin(Div(Mul(Pi, r), n))))))),
    Variables(r, n),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(r, Range(1, Sub(n, 1))))))

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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