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Fungrim entry: bed7ee

ζ ⁣(s,a+N)=ζ ⁣(s,a)n=0N11(n+a)s\zeta\!\left(s, a + N\right) = \zeta\!\left(s, a\right) - \sum_{n=0}^{N - 1} \frac{1}{{\left(n + a\right)}^{s}}
Assumptions:sC  and  aC  and  s1  and  (a{0,1,}  or  Re(s)<0  or  s=0)  and  NZ1s \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) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
\zeta\!\left(s, a + N\right) = \zeta\!\left(s, a\right) - \sum_{n=0}^{N - 1} \frac{1}{{\left(n + a\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 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(HurwitzZeta(s, Add(a, N)), Sub(HurwitzZeta(s, a), Sum(Div(1, Pow(Add(n, a), s)), For(n, 0, Sub(N, 1)))))),
    Variables(s, a, N),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)), Element(N, ZZGreaterEqual(1)))))

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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