Aj. Macleod, HIGH-ACCURACY NUMERICAL VALUES IN THE GAUSS-KUZMIN CONTINUED-FRACTIONPROBLEM, Computers & mathematics with applications, 26(3), 1993, pp. 37-44
If xepsilon[0, 1], we can expand x in a regular continued fraction x =
1/(a1 + 1/(a2 + 1/(a3 + ...))). The integers a(i), i = 1,... can be g
enerated by the recursion a(i) = int (1/x(i-1)), x(i) = 1/x(i-1) - a(i
), i = 1,..., with x0 = x. If x is uniformly distributed in [0, 1], de
fine F(n)(x) = Prob (x(n) < x). Babenko showed F(n)(x) = ln(1 + x)/ln
2 + SIGMA(k=2)infinity lambda(k)(n)B(k)(x). In this work, we report ve
ry high accuracy values for lambda2 to lambda10, together with 10D Che
byshev coefficients for B2 to B-10. Simulation results suggest that th
ese values give very accurate approximations.