Formula:DLMF:25.11:E14

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E15 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{-n}{a}=-\frac{\BernoulliB{n+1}@{a}}{n+1}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (14), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E15 >>