Formula:DLMF:25.6:E13

<< Formula:DLMF:25.6:E12

formula in Integer Arguments

Formula:DLMF:25.6:E14 >>

${\displaystyle{\displaystyle{\displaystyle(-1)^{k}\RiemannZeta^{(k)}@{-2n}=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im{(c^{k-m})}\*{\Gamma^{(r)}}\left(2n+1\right)\RiemannZeta^{(m-r)}@{2n+1}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\displaystyle c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Im{z}}}}$ : imaginary part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation (13), Section 25.6 of DLMF.

URL links

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