A Gauss-Kuzmin Theorem and Related Questions forθ-Expansions

Abstract

Using the natural extension for θ-expansions, we give an infiniteorder-chain representation of the sequence of the incomplete quotients of these expansions. Together with the ergodic behavior of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion. (2010). Mathematics Subject Classifications

Computation of the Best Approximations for θ-Expansion of Continued Fractions using Maple

“…Suppose that throughout our discussion, we referred our expansion as the continued fraction expansion of a number in (0, ). One expansion discussed by (Chakraborty & Rao, 2003), which was studied in detail by (Sebe & Lascu, 2014), had raised a different type of continued fractions, namely -expansions. (Chakraborty & Rao, 2003) and it is called the infinite or non-terminating continued fraction expansion of with respect to .…”

On convergence rate in the Gauss–Kuzmin problem for θ-expansions

“…then the sequence (a n ) n∈N + in (2.1) is obtained as follows: This new expansion of positive reals, different from the regular continued fraction expansion, was also studied in [2,8,12,13].…”

“…The Perron-Frobenius operator of ([0, θ], B [0,θ] , γ θ , T θ ) is defined as the bounded linear operator U on the Banach space L 1 ([0, θ], γ θ ) such that the following holds [13]:…”

On the metrical theory of a non-regular continued fraction expansion

“…The Gauss-Kuzmin-Lévy theorem is the first basic result in the rich metrical theory of continued fractions. Generalizations of these problems for nonregular continued fractions are also called as the Gauss-Kuzmin problem and the Gauss-Kuzmin-Lévy problem [7,10,12,13,14].…”