Formula:DLMF:25.14:E4

<< Formula:DLMF:25.14:E3

formula in Lerch's Transcendent

Formula:DLMF:25.14:E5 >>

${\displaystyle{\displaystyle{\displaystyle\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a\neq 0,-1,-2,\dots,|z|<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{s}>1,|z|=1}}}$ &
${\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Phi}}}$ : Lerch's transcendent : http://dlmf.nist.gov/25.14#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (4), Section 25.14 of DLMF.

URL links

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<< Formula:DLMF:25.14:E3

formula in Lerch's Transcendent

Formula:DLMF:25.14:E5 >>