Symmetry in the sequence of approximation coefficients

TL;DR

This paper uncovers a symmetry in the sequences of partial quotients and approximation coefficients of continued fractions, providing functions to recover these sequences and revealing their elegant interrelations.

Contribution

It introduces a new function linking consecutive approximation coefficients and partial quotients, demonstrating a symmetry that enables sequence reconstruction from limited data.

Findings

Established a function relating $a_{n+1}$ to $ heta_{n\u00b1}$ and $ heta_n$

Expressed $ heta_{n1}$ as a function of $( heta_{n1}, heta_n)$

Revealed an elegant symmetry in the sequence of approximation coefficients

Abstract

Let $\{a_n\}_1^\infty$ and $\{\theta_n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $f$ such that $a_{n+1} = f(\theta_{n\pm1},\theta_n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $\theta_{n \pm 1}$ as a function of $(\theta_{n \mp 1}, \theta_n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.