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

TL;DR

This paper investigates the asymptotic behavior of distribution functions in odd continued fraction expansions using Markov chain analysis and Sz"usz's method, providing insights into convergence rates and solution approaches.

Contribution

It introduces multiple approaches, including Markov chain transition operators and Sz"usz's method, to analyze the asymptotics of odd continued fractions.

Findings

Established asymptotic distribution functions for odd continued fractions.

Provided estimates for convergence rates in the Gauss-Kuzmin-type problem.

Compared different analytical methods for studying these expansions.

Abstract

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.