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arXiv · 1903.00419

Hecke groups, linear recurrences, and Kepler limits

Barry Brent
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Abstract

We study the linear fractional transformations in the Hecke group $G(Φ)$ where $Φ$ is either root of $x^2 - x -1$ (the larger root being the "golden ratio" $ϕ= 2 \cos \frac π5$.) Let $g \in G(Φ)$ and let $z$ be a generic element of the upper half-plane. Exploiting the fact that $Φ^2 = Φ-1$, 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 $Φ$ with coefficients that are certain multiples of Fibonacci numbers. We make somewhat less detailed observations along similar lines about the functions in $G(2 \cos \frac πk)$ for $k \geq 5$.

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Barry Brent. 2021-02-18. Hecke groups, linear recurrences, and Kepler limits. https://arxiv.org/abs/1903.00419

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