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Fungrim entry: ed4f6f

ζ ⁣(s,a+1)=ζ ⁣(s,a)1as\zeta\!\left(s, a + 1\right) = \zeta\!\left(s, a\right) - \frac{1}{{a}^{s}}
Assumptions:sC  and  aC  and  s1  and  (a{0,1,}  or  Re(s)<0  or  s=0)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)
\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)
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
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
Source code for this entry:
    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)))))

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2021-03-15 19:12:00.328586 UTC