Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

Steven J. Miller,Daniel Fishman

doi:10.1155/2013/414623

Steven J. Miller, Daniel Fishman

Open Access

https://doi.org/10.1155/2013/414623

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Abstract

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.

Highlights

In [1], van der Poorten wrote that the elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. is makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio
As our work is an outgrowth of [1], we happily refer the reader to that paper for some basic background information on continued fractions, and to the books [2, 3] for proofs
We explore the continued fraction expansions of powers of quadratic surds

Summary

Introduction

In [1], van der Poorten wrote that the elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. is makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio. He derived expansions for the continued fraction of √DD and (1+ √DDDDD (with DD D DDDD D) and for the expansions of some simple functions of these numbers as well as numbers related to Diophantine equations similar to Pell’s equation Another technique that shows promise in manipulating continued fractions comes from an un nished paper of Gosper [4]. Is operation is carried out using two-dimensional arrays in the simple cases and requires added dimensions when considering functions of two or more quadratic irrationals These algorithms, while useful for many applications, do not reveal the nature of the underlying structure in a closed form in an accessible manner. Alf ’s constant enthusiasm, knowledge of the eld, and helpful comments greatly improved the exposition of continued fractions in [3], a textbook developed from the Princeton course

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