We introduced a new continued fraction expansions in our previous paper. For these expansions, we show the Brodén-Borel-Lévy type formula. Furthermore, we compute the transition probability function from this and the symbolic dynamical system of the natural number with the unilateral shift.
 Adams, W.W. and Davison, J.L., A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977) 194-198.
 Boyarsky, A. and Góra, P., Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Birkhäuser, Boston, 1997.
 Brezinski, C., History of Continued Fractions and Padé Approximants. Springer Series in Computational Mathematics 12, Springer-Verlag, Berlin, 1991.
 Corless, R.M., Continued fractions and chaos, Amer. Math. Monthly 99(3) (1992), 203-215.
 Davison, J.L., A series and its associated continued fraction, Proc. Amer.Math. Soc.63(1) (1977) 29-32.
 Iosifescu, M. and Kraaikamp C., Metrical theory of continued fractions, Kluwer Academic, 2002.
 Iosifescu, M. and Sebe, G.I., An exact convergence rate in a Gauss- Kuzmin-Lévy problem for some continued fraction expansion, in vol. Mathematical Analysis and Applications, 90-109. AIP Conf. Proc. 835 (2006), Amer.Inst.Physics, Melville, NY.
 Kuzmin, R.O., On a problem of Gauss. Dokl. Akad. Nauk SSSR Ser. A (1928) 375-380. [Russian; French version in Atti Congr. Internaz.Mat. (Bologna, 1928), Tomo VI, pp.83-89. Zanichelli, Bologna, 1932].
 Lascu, D., Markov processes in probabilistic number theory, Ph.D. thesis, Romanian Academy, 2010. (Romanian)
 Lascu, D., On a Gauss-Kuzmin-type problem for a family of continued fraction expansions, J. Number Theory 133(7) (2013), 2153-2181.
 Lévy, P., Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue. Bull. Soc. Math. France 57 (1929) 178-194.
 Sebe, G.I., On convergence rate in the Gauss-Kuzmin problem for grotesque continued fractions, Monatsh. Math. 133 (2001), 241-254.
 Sebe, G.I., A Gauss-Kuzmin theorem for the Rosen fractions, J.Théor.Nombres Bordx. 14 (2002), 667-682.
 Sebe, G.I. and Lascu, D. A Gauss-Kuzmin theorem and related questions for theta-expansions, arXiv preprint arXiv:1305.5563 (2013), 1-27.