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)?
2026-03-25 06:04:36.1774418676
Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$
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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: