Formula:DLMF:25.11:E10

<< Formula:DLMF:25.11:E9

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E11 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{\infty}\frac{\Gamma\left(n+s\right)}{n!\Gamma\left(s\right)}\RiemannZeta@{n+s}(1-a)^{n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle|a-1|<1}}}$

Proof

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Use Taylor's theorem and

${\displaystyle{\displaystyle{\displaystyle\frac{\partial}{\partial a}\HurwitzZeta@{s}{a}=-s\HurwitzZeta@{s+1}{a}}}}$.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (10), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E9

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E11 >>