High-accuracy numerical values in the Gauss-Kuzmin continued fraction problem (Q2367557)

If \(x\in [0,1]\), we can expand \(x\) in a regular continued fraction \(x= 1/(a_ 1+ 1/(a_ 2+ 1/(a_ 3+ \dots)))\). The integers \(a_ i\), \(i=1,\dots\) can be generated by the recursion \(a_ i= \text{int} (1/x_{i-1})\), \(x_ i= 1/x_{i-1}- a_ i\), \(i=1,\dots\), with \(x_ 0=x\). If \(x\) is uniformly distributed in \([0,1]\), define \(F_ n(x)= \text{Prob}(x_ n<x)\). \textit{K. I. Babenko} [Sov. Math., Dokl. 19, 136- 140 (1978); translation from Dokl. Akad. Nauk SSSR 238, 1021-1024 (1978; Zbl 0389.10036); see also \textit{K. I. Babenko} and \textit{S. P. Yur'ev}, Sov. Math., Dokl. 19, 731-735 (1978); translation from Dokl. Akad. Nauk SSSR 240, 1273-1276 (1978; Zbl 0416.10040)] showed \[ F_ n(x)= {{\ln(1+x)} \over {\ln 2}}+ \sum_{k=2}^ \infty \lambda_ k^ n B_ k(x). \] In this work, we report very high accuracy values for \(\lambda_ 2\) to \(\lambda_{10}\), together with \(10D\) Chebyshev coefficients for \(B_ 2\) to \(B_{10}\). Simulation results suggest that these values give very accurate approximations.