where
(OEIS A038517; Knuth 1998, p. 350) and
is an analytic function with .
was computed to about 30 decimal places by Flajolet and Vallée (1995) and
to 100 places by Sebah (unpublished). Briggs (2003) computed as the negative of the second largest (in absolute value)
eigenvalue of the matrix defined by
Babenko, K. I. "On a Problem of Gauss." Soviet Math. Dokl.19, 136-140, 1978.Bailey, D. H.; Borwein,
J. M.; and Crandall, R. E. "On the Khintchine Constant." Math.
Comput.66, 417-431, 1997.Briggs, K. "A Precise Computation
of the Gauss-Kuzmin-Wirsing Constant." Preliminary report. 2003 July 8. http://keithbriggs.info/documents/wirsing.pdf.Daudé,
H.; Flajolet, P.; and Vallé, B. "An Average-Case Analysis of the Gaussian
Algorithm for Lattice Reduction." Combin. Probab. Comput.6, 397-433,
1997.Finch, S. R. "Gauss-Kuzmin-Wirsing Constant." §2.17
in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 151-156,
2003.Flajolet, P. and Vallée, B. "On the Gauss-Kuzmin-Wirsing
Constant." Unpublished memo. 1995. http://algo.inria.fr/flajolet/Publications/gauss-kuzmin.ps.Knuth,
D. E. The
Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, p. 341, 1998.MacLeod, A. J. "High-Accuracy
Numerical Values of the Gauss-Kuzmin Continued Fraction Problem." Computers
Math. Appl.26, 37-44, 1993.Mayer, D. H. "Continued
Fractions and Related Transformations." In Ergodic
Theory, Symbolic Dynamics and Hyperbolic Spaces. Papers from the Workshop on Hyperbolic
Geometry and Ergodic Theory held in Trieste, April 17-28, 1989 (Ed. T. Bedford,
M. Keane, and C. Series). New York: Clarendon Press, pp. 175-222,
1991.Plouffe, S. "The Gauss-Kuzmin-Wirsing Constant." http://pi.lacim.uqam.ca/piDATA/gkw.txt.Sloane,
N. J. A. Sequences A007515/M2267
and A038517 in "The On-Line Encyclopedia
of Integer Sequences."Wirsing, E. "On the Theorem of Gauss-Kuzmin-Lévy
and a Frobenius-Type Theorem for Function Spaces." Acta Arith.24,
507-528, 1974.
is the Gauss-Kuzmin
distribution, then
(OEIS A038517; Knuth 1998, p. 350) and
is an analytic function with 
.
was computed to about 30 decimal places by Flajolet and Vallée (1995) and
to 100 places by Sebah (unpublished). Briggs (2003) computed 
as the negative of the second largest (in absolute value)
eigenvalue of the 
matrix defined by
![M_(jk)=((-1)^j)/(j!(-2)^k)sum_(i=0)^k(k; i)(-2)^i(i+2)_j[zeta(i+j+2)(2^(i+j+2)-1)-2^(i+j+2)]](/images/equations/Gauss-Kuzmin-WirsingConstant/NumberedEquation2.svg)
,
where 
is a binomial coefficient, 
is a Pochhammer symbol,
and 
is the Riemann zeta function. For example,
![M_2=[1/2(pi^2-8) 7zeta(3)-1/4pi^2-6; 16-14zeta(3) 7zeta(3)-1/2pi^4+40].](/images/equations/Gauss-Kuzmin-WirsingConstant/NumberedEquation3.svg)
and a precision of 1300 bits to obtain 385 digits.
