# Fungrim entry: 34fafa

$\psi^{(m)}\!\left(z + n\right) = \psi^{(m)}\!\left(z\right) + {\left(-1\right)}^{m} m ! \sum_{k=0}^{n - 1} \frac{1}{{\left(z + k\right)}^{m + 1}}$
Assumptions:$m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\psi^{(m)}\!\left(z + n\right) = \psi^{(m)}\!\left(z\right) + {\left(-1\right)}^{m} m ! \sum_{k=0}^{n - 1} \frac{1}{{\left(z + k\right)}^{m + 1}}

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
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("34fafa"),
Formula(Equal(DigammaFunction(Add(z, n), m), Add(DigammaFunction(z, m), Mul(Mul(Pow(-1, m), Factorial(m)), Sum(Div(1, Pow(Add(z, k), Add(m, 1))), For(k, 0, Sub(n, 1))))))),
Variables(m, z, n),
Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)), Element(n, ZZGreaterEqual(0)))))

## Topics using this entry

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