Fungrim entry: 3fe553

\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)

Assumptions:

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\}

TeX:

\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`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("3fe553"),
    Formula(Equal(DigammaFunction(Div(p, q)), Add(Sub(Sub(Neg(ConstGamma), Log(Mul(2, q))), Mul(Div(Pi, 2), Cot(Div(Mul(Pi, p), q)))), Mul(2, Sum(Mul(Cos(Div(Mul(Mul(Mul(2, Pi), k), p), q)), Log(Sin(Div(Mul(Pi, k), q)))), For(k, 1, Floor(Div(Sub(q, 1), 2)))))))),
    Variables(p, q),
    Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(p, Range(1, Sub(q, 1))))))

Fungrim entry: 3fe553

Topics using this entry