Abstract

We explore a new form of periodic behavior among continued fractions having telescoping periods. Informally speaking, a quasiperiodic continued fraction $\xi ={[a_0,a_1,\dots <!--FUNCTION\; APPLICATION-->,a_{\rho -1},\overline{f_0(k),f_1(k),\dots <!--FUNCTION\; APPLICATION-->,f_{\omega -1}(k)}]}_{k=1}^{\infty }$ is stretch-periodic if the period block of $\xi$ contains one or more substrings of partial quotients increasing in length as $k$ increases. We explore three main ideas. First, we will formalize stretch-periodic continued fractions and show that they can be characterized as a special quasiperiodic continued fraction. We achieve this by introducing a class of such fractions having regular classical quasiperiodic representations where the partial quotients under the period block can be expressed in terms of floor and ceiling functions of order $\mathcal{𝒪}(n^2)$. Secondly, we give an historical example of the stretch-periodic continued fraction after applying a linear fractional transformation to Hall’s number $\mathcal{\mathscr{H}}$. Finally, we then identify how stretch-periodicity occurs naturally through the least integer algorithm giving continued fractions of the form $\xi ={[a_0,a_1,a_2,\dots <!--FUNCTION\; APPLICATION-->]}^{-}$, called least integer continued fractions (LICFs), where

We then take a closer look into negative unary continued fractions (NUCFs), and present a theorem showing that by deviating from the least integer algorithm, every irrational number has uncountably many NUCF representations.

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