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Symmetry in the sequence of approximation coefficients

Author: Avraham Bourla
Journal: Proc. Amer. Math. Soc. 141 (2013), 3681-3688
MSC (2010): Primary 11A55, 11J70, 11J71, 11J81; Secondary 01-02
Published electronically: July 17, 2013
MathSciNet review: 3091760
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

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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

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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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