Formula:DLMF:25.16:E11

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Formula:DLMF:25.16:E12 >>

${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{n}\frac{1}{m^{z}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{(s+z)}>1}}}$

Note(s)

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}}}}$ is the special case ${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{1}}}}$ of ${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle H}}}$ : generalized Euler sums : http://dlmf.nist.gov/25.16#SS2.p7
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle H}}}$ : Euler sums : http://dlmf.nist.gov/25.16#SS2.p1

Bibliography

Equation (11), Section 25.16 of DLMF.

URL links

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