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Fungrim entry: 95e270

ζ(r) ⁣(s,a+N)=ζ(r) ⁣(s,a)+(1)r+1k=0N1logr ⁣(a+k)(a+k)s\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:sC  and  aC  and  s1  and  (a{0,1,}  or  Re(s)<0)  and  NZ1  and  rZ0s \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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
Loglog(z)\log(z) Natural logarithm
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:
Entry(ID("95e270"),
    Formula(Equal(HurwitzZeta(s, Add(a, N), r), Add(HurwitzZeta(s, a, r), Mul(Pow(-1, Add(r, 1)), Sum(Div(Pow(Log(Add(a, k)), r), Pow(Add(a, k), s)), For(k, 0, Sub(N, 1))))))),
    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)))))

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2020-08-27 09:56:25.682319 UTC