# Fungrim entry: 77e507

$\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
Log$\log(z)$ Natural logarithm
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("77e507"),
Formula(Equal(HurwitzZeta(s, a, r), Mul(Pow(-1, r), Sum(Div(Pow(Log(Add(n, a)), r), Pow(Add(n, a), s)), For(n, 0, Infinity))))),
Variables(s, a, r),
Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(r, 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.

2021-03-15 19:12:00.328586 UTC