On a tree of rational functions related to continued fractions

TL;DR

This paper introduces a binary tree of rational functions inspired by continued fractions, revealing properties about the roots of denominators, density of functions, and connections to Farey trees and backward continued fractions.

Contribution

It constructs a novel binary tree of rational functions related to continued fractions and explores its properties, including roots, density, and connections to classical Farey trees.

Findings

Denominators' zeros are real negative and dense in (-∞, -1]

Functions for x>0 are non-intersecting and dense in (0,1)

Tree contains all rationals in (0,1) exactly once for integer x

Abstract

In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example, the zero solutions of the denominators $q(x)$ are all real negative numbers and are dense in $(-\infty,-1]$. For $x>0$ functions are non intersecting and form a dense subset of $(0,1)$. Furthermore, when evaluating the tree for positive rational values, the tree contains every rational in $(0,1)$ exactly once if and only if $x\in \mathbb{N}$. For $x=1$, one finds back the classical Farey tree which is related to regular continued fractions. In the last part, we will make a similar tree in a similar way but for backward continued fractions. We highlight some similarities and differences.