Fungrim entry: 739819

\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:

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
`Sum`	$\sum_{n} f(n)$	Sum
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Log`	$\log(z)$	Natural logarithm
`Sin`	$\sin(z)$	Sine
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	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))))))

Topics using this entry

Digamma function