# Fungrim entry: ed4f6f

$\zeta\!\left(s, a + 1\right) = \zeta\!\left(s, a\right) - \frac{1}{{a}^{s}}$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)$
TeX:
\zeta\!\left(s, a + 1\right) = \zeta\!\left(s, a\right) - \frac{1}{{a}^{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 \;\mathbin{\operatorname{or}}\; s = 0\right)
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("ed4f6f"),
Formula(Equal(HurwitzZeta(s, Add(a, 1)), Sub(HurwitzZeta(s, a), Div(1, Pow(a, s))))),
Variables(s, a),
Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 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