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Fungrim entry: 77e507

ζ(r) ⁣(s,a)=(1)rn=0logr ⁣(n+a)(n+a)s\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}
Assumptions:sC  and  Re(s)>1  and  aC{0,1,}  and  rZ0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}
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
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(HurwitzZeta(s, a, r), Mul(Pow(-1, r), Sum(Div(Pow(Log(Add(n, a)), r), Pow(Add(n, a), s)), For(n, 0, Infinity))))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(r, ZZGreaterEqual(0)))))

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2021-03-15 19:12:00.328586 UTC