Continued Fractions and Linear Fractional Transformations

TL;DR

This paper explores the relationship between continued fractions and linear fractional transformations, identifying conditions for their convergence and connections to Pell equation solutions.

Contribution

It establishes precise conditions under which linear fractional transformations produce continued fraction convergents and describes the structure of these continued fractions.

Findings

Convergents coincide infinitely often with continued fraction convergents when a specific integrality condition holds.

The continued fraction structure can be described as concatenations of fractions of rational numbers.

The orbit of the transformation often includes solutions to Pell equations.

Abstract

Rational approximations to a square root $\sqrt{k}$ can be produced by iterating the transformation $f(x) = (dx+k)/(x+d)$ starting from $\infty$ for any positive integer $d$. We show that these approximations coincide infinitely often with continued fraction convergents if and only if $4d^2/(k-d^2)$ is an integer, in which case the continued fraction has a rich structure. It consists of the concatenation of the continued fractions of certain explicitly definable rational numbers, and it belongs to one of infinitely many families of continued fractions whose terms vary linearly in two parameters. We also give conditions under which the orbit $\{f^n(\infty)\}$ consists exclusively of convergents or semiconvergents and prove that with few exceptions it includes all solutions $p/q$ to the Pell equation $p^2 - k q^2 = \pm 1$.