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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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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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  • $\begingroup$ Even better, see the "Lambert series" (7) on the same page. $\endgroup$
    – GEdgar
    Sep 27, 2012 at 14:44
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    $\begingroup$ @GEdgar I'm not sure the Lambert series is enough to write the OP's function for every $z.$ $\endgroup$
    – Ragib Zaman
    Sep 27, 2012 at 14:53
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Mathematica gives

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

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