Fungrim entry: 7b724b

\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)

Assumptions:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}

TeX:

\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("7b724b"),
    Formula(Equal(DigammaFunction(Mul(n, z), m), Mul(Div(1, Pow(n, Add(m, 1))), Sum(DigammaFunction(Add(z, Div(k, n)), m), For(k, 0, Sub(n, 1)))))),
    Variables(m, n, z),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)), Element(z, CC), NotElement(Mul(n, z), ZZLessEqual(0)))))

Topics using this entry

Digamma function