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Fungrim entry: 6c3523

ζ ⁣(s,12+n)=(2s1)ζ ⁣(s)2sk=0n11(2k+1)s\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}
Assumptions:sC  and  nZ0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
TeX:
\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Powab{a}^{b} Power
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Sumnf(n)\sum_{n} f(n) Sum
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("6c3523"),
    Formula(Equal(HurwitzZeta(s, Add(Div(1, 2), n)), Sub(Mul(Sub(Pow(2, s), 1), RiemannZeta(s)), Mul(Pow(2, s), Sum(Div(1, Pow(Add(Mul(2, k), 1), s)), For(k, 0, Sub(n, 1))))))),
    Variables(s, n),
    Assumptions(And(Element(s, CC), Element(n, ZZGreaterEqual(0)))))

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2020-08-27 09:56:25.682319 UTC