Symmetry in the sequence of approximation coefficients

Avraham Bourla

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arXiv · 1110.3005

Symmetry in the sequence of approximation coefficients

Avraham Bourla

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Abstract

Let $\{a_n\}_1^\infty$ and $\{θ_n\}_0^\infty$ 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 $a_{n+1} = f(θ_{n\pm1},θ_n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $θ_{n \pm 1}$ as a function of $(θ_{n \mp 1}, θ_n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.

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Avraham Bourla. 2013-04-19. Symmetry in the sequence of approximation coefficients. https://arxiv.org/abs/1110.3005

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