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I'd like to have a more compact form for


Could anyone devise something?


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Similar question:… – Gerry Myerson Sep 27 '12 at 12:18
up vote 4 down vote accepted

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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Even better, see the "Lambert series" (7) on the same page. – GEdgar Sep 27 '12 at 14:44
@GEdgar I'm not sure the Lambert series is enough to write the OP's function for every $z.$ – Ragib Zaman Sep 27 '12 at 14:53

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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