# Hecke groups, linear recurrences, and Kepler limits

## TL;DR

This paper explores the properties of linear fractional transformations in Hecke groups related to the golden ratio and other cosine-based groups, revealing connections with Fibonacci numbers and linear recurrences.

## Contribution

It provides explicit formulas for transformations in Hecke groups involving Fibonacci numbers and extends observations to groups generated by cosines of rational multiples of pi.

## Key findings

- Transformations are quotients of linear polynomials in z with coefficients linked to Fibonacci numbers.
- Coefficients in these transformations are linear in the golden ratio and Fibonacci multiples.
- Observations extend to functions in groups generated by cosines of b5k for k 5 5.

## Abstract

We study the linear fractional transformations in the Hecke group $G(\Phi)$ where $\Phi$ is either root of $x^2 - x -1$ (the larger root being the "golden ratio" $\phi = 2 \cos \frac {\pi}5$.) Let $g \in G(\Phi)$ and let $z$ be a generic element of the upper half-plane. Exploiting the fact that $\Phi^2 = \Phi -1$, we find that $g(z)$ is a quotient of linear polynomials in $z$ such that the coefficients of $z^1$ and $z^0$ in the numerator and denominator of $g(z)$ appear themselves to be linear polynomials in $\Phi$ with coefficients that are certain multiples of Fibonacci numbers. We make somewhat less detailed observations along similar lines about the functions in $G(2 \cos \frac {\pi}k)$ for $k \geq 5$.