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I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured $$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{...

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...

Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form: $\operatorname{li}^{-...