# Fungrim entry: 95e270

$\zeta^{(r)}\!\left(s, a + N\right) = \zeta^{(r)}\!\left(s, a\right) + {\left(-1\right)}^{r + 1} \sum_{k=0}^{N - 1} \frac{\log^{r}\!\left(a + k\right)}{{\left(a + k\right)}^{s}}$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
\zeta^{(r)}\!\left(s, a + N\right) = \zeta^{(r)}\!\left(s, a\right) + {\left(-1\right)}^{r + 1} \sum_{k=0}^{N - 1} \frac{\log^{r}\!\left(a + k\right)}{{\left(a + k\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\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
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Re$\operatorname{Re}(z)$ Real part
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("95e270"),
Variables(s, a, N, r),
Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0)), Element(N, ZZGreaterEqual(1)), 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