Fungrim home page

# Fungrim entry: c687d6

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

m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z - n \notin \{0, -1, \ldots\}
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("c687d6"),
Formula(Equal(DigammaFunction(Sub(z, n), m), Sub(DigammaFunction(z, m), Mul(Mul(Pow(-1, m), Factorial(m)), Sum(Div(1, Pow(Sub(z, k), Add(m, 1))), For(k, 1, n)))))),
Variables(m, z, n),
Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), Element(n, ZZGreaterEqual(0)), NotElement(Sub(z, n), ZZLessEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC