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I'd like to have a more compact form for

$f(z)=\sum_{k=0}^{\infty}\frac{1}{2^{z+k}-1}$

Could anyone devise something?

Thanks.

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    Similar question: http://math.stackexchange.com/questions/203125/can-this-series-be-expressed-in-closed-form-and-if-so-what-is-it2012-09-27

2 Answers 2

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Your function can be represented in terms (and is actually quite close to the definition) of the q-Polygamma function. See equation (2).

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    @GEdgar I'm not sure the Lambert series is enough to write the OP's function for every $z.$2012-09-27
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Mathematica gives

$f(z)=-\frac{\text{Log}[2]+2 \text{Log}\left[2^{-z}\right]+2 \text{QPolyGamma}\left[0,-\frac{\text{Log}\left[2^{-z}\right]}{\text{Log}[2]},2\right]}{\text{Log}[4]}$