Formula:DLMF:25.11:E21

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E22 >>

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HurwitzZeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n} - \ln@{2 \cpi k}) \BernoulliB{2n}@{h/k}}{2n} - \frac{(\digamma@{2n} - \ln@{2 \cpi}) \BernoulliB{2n}}{2 n k^{2n}} + \frac{\opminus^{n+1} \cpi}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \sin@{\frac{2 \cpi r h}{k}} \digamma^{(2n-1)}@{\frac{r}{k}} + \frac{\opminus^{n+1} 2 \cdot (2n-1)!}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \cos@{\frac{2 \cpi r h}{k}} \HurwitzZeta'@{2n}{\frac{r}{k}} + \frac{\RiemannZeta'@{1-2n}}{k^{2n}} }}

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle h,k}}}$ are integers with ${\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}}$ &
${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (21), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E20

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E22 >>