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I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet for an hour did not give any results. Could you please remind me this identity (if it exists)?

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    $\begingroup$ Are you sure about $1/4$ and not $1/3$? I am looking at (5) and (7) here; recall that Clausen function is essentially imaginary part of the dilogarithm... $\endgroup$
    – Start wearing purple
    Commented Aug 10, 2014 at 23:02
  • $\begingroup$ Definitely $1/4$. IIRC, there was a difference of polygammas of variable order $\nu$ at points $3/4$ and $1/4$ expressed via $\sum$ of polylogarithms depending on $\nu$. $\endgroup$
    – Vladimir Reshetnikov
    Commented Aug 10, 2014 at 23:26
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    $\begingroup$ Closest I can find is the reflection relation for the polygamma function: $$\psi_n(1-z)-(-1)^n \psi_n(z)=(-1)^n \pi \dfrac{d^n}{dz^n}\cot \pi x.$$ Maybe a special case of that can be written in terms of polylogs? $\endgroup$
    – Semiclassical
    Commented Aug 11, 2014 at 2:10

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I have not found a source, but I think I reconstructed it correctly: $$\psi^{(n)}\!\left(\tfrac34\right)-\psi^{(n)}\!\left(\tfrac14\right)=(-1)^n\,4^{n+1}\,n!\,\,\Im\operatorname{Li}_{n+1}(i),\ n\in\mathbb N.$$ In the source where I saw it, the imaginary part was probably written as a difference of two polylog terms.

It can be proved using:

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