Fungrim entry: 40c3e2

\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`CC`	$\mathbb{C}$	Complex numbers
`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("40c3e2"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(a, a, r)), Mul(RisingFactorial(Sub(Sub(1, s), r), r), HurwitzZeta(Add(s, r), a)))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), NotEqual(Add(s, r), 1), Element(a, CC), Greater(Re(a), 0), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Hurwitz zeta function