Some asymptotic results for the continued fraction expansions with odd partial quotients

We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these expansions on a certain Banach space of complex-valued functions of bounded variation we make a brief survey of the solution in the Gauss-Kuzmin-type problem. Secondly, we use the method of Sz\"usz to obtain a similar asymptotic result and to give a good estimate of the convergence rate involved.