On the geometry behind a recurrent relation

TL;DR

This paper explores the algebraic and geometric properties of a specific linear recursive relation to estimate the number of valence chains in Farey series, linking recursive relations with geometric structures.

Contribution

It introduces a novel approach connecting recursive relations with geometric analysis to estimate valence chains in Farey series.

Findings

Derived bounds for the number of valence chains

Established geometric interpretations of the recursive relation

Provided algebraic insights into Farey series structures

Abstract

We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.