Fungrim entry: 95e270

\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:

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}

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
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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)))))

Topics using this entry

Hurwitz zeta function