Symmetry in the sequence of approximation coefficients

Details Symmetry in the sequence of approximation coefficients Symmetry in the sequence of approximation coefficients

2011-10-13

arxiv.org/abs/1110.3005v7

Mathematics

Number Theory

Scientific paper

Let $\{a_n\}_1^\infty$ and $\{\theta_n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the regular continued fraction expansion of an irrational number in the unit interval. We are going to find a real valued function $f$ on two variables, such that $a_{n+1} = f(\theta_{n - 1},\theta_n) = f(\theta_{n+1},\theta_n)$. In tandem with a formula due to Dajani and Kraainkamp \cite{DK}, we will write $\theta_{n \pm 1}$ in terms of $(\theta_{n \mp 1}, \theta_n)$, revealing an elegant symmetrical structure in this classical sequence. In particular, this will enable us to recover the entire sequence from a pair of consecutive terms.

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