Shrinking the period lengths of continued fractions while still capturing convergents

Abstract

Here we prove that every real quadratic irrational α can be expressed as a periodic non-simple continued fraction having period length one. Moreover, we show that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of α. We close with an application relating the structure of a quadratic α to its conjugate.

On Periodic P-Continued Fraction Having Period Length One

“…We establish our result by extending some of the arguments of [1], in this new context and making appropriate adjustments.…”

“…In [1] Burger et al prove that every real quadratic irrational α can be expressed as a periodic non-simple continued fraction having period length one. Moreover, it is proved that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of α, For further references on the subject, see also [3], [2] and [4].…”

Dependence with complete connections and the Gauss–Kuzmin theorem for N-continued fractions

“…The purpose of this paper is to prove a GaussKuzmin type problem for Ncontinued fraction expansions introduced by Burger et al [3]. In order to solve the problem, we apply the theory of random systems with complete connections by Iosifescu [9].…”

“…}, with T 0 N (x) = x. By the very definitions, Burger et al proved in [3] that any irrational 0 < x < 1 can be written in the form where a n 's are non-negative integers. We will call (1.…”

Continued fraction expansions with variable numerators

“…More recently, Edward Burger and his co-authors showed in [3] that a similar result also holds if a n = N for all n ≥ 1 and for infinitely many positive integers N . However, in their result, the continued fraction expansion (which is now a so-called "N -expansion") is not necessarily proper, i.e., we need not have that b n ≥ N for all n ≥ 1.…”