Symmetry in the sequence of approximation coefficients
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- by Avraham Bourla PDF
- Proc. Amer. Math. Soc. 141 (2013), 3681-3688 Request permission
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 $a_{n+1} = f(\theta _{n\pm 1},\theta _n)$. 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.References
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Additional Information
- Avraham Bourla
- Affiliation: Department of Mathematics. Saint Mary’s College of Maryland, Saint Mary’s City, Maryland 20686
- Address at time of publication: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016
- Email: abourla@smcm.edu, bourla@american.edu
- Received by editor(s): October 16, 2011
- Received by editor(s) in revised form: December 1, 2011, and December 8, 2011
- Published electronically: July 17, 2013
- Communicated by: Matthew A. Papanikolas
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3681-3688
- MSC (2010): Primary 11A55, 11J70, 11J71, 11J81; Secondary 01-02
- DOI: https://doi.org/10.1090/S0002-9939-2013-11601-1
- MathSciNet review: 3091760