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Fungrim entry: 40c3e2

drdarζ ⁣(s,a)=(1sr)rζ ⁣(s+r,a)\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)
Assumptions:sC  and  s1  and  s+r1  and  aC  and  Re(a)>0  and  rZ0s \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}
\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}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    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)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC