Formula:DLMF:25.11:E38

<< Formula:DLMF:25.11:E37

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E39 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\binom{n+k}{k}\zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a\right)-{\psi^{(n)}}\left(a-z\right)\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle|z|<|a|}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (38), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E37

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E39 >>