%0 Journal Article
%T Closed Form Continued Fraction Expansions of Special Quadratic Irrationals
%A Daniel Fishman
%A Steven J. Miller
%J ISRN Combinatorics
%D 2013
%R 10.1155/2013/414623
%X 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. 1. 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. This 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. Briefly, every real number has a continued fraction expansion where each is an integer (and a positive integer unless ). The ¡¯s are called the partial quotients. For brevity we often write If we truncate the expansion at , we obtain the th partial quotient The ¡¯s and ¡¯s satisfy the very important relation Continued fractions encode much useful information about the algebraic structure of a number and frequently arise in approximation theory and dynamical systems. Clearly, is rational if and only if its continued fraction is finite, and a beautiful theorem of Lagrange states that is a quadratic irrational if and only if the continued fraction expansion is periodic. In this paper, we explore the continued fraction expansions of powers of quadratic surds. Recall that a quadratic surd is an irrational number of the form , where and is a nonsquare integer. By Lagrange¡¯s theorem we know these numbers and their powers have periodic continued fractions, which suggests many questions, such as what is the length of the period as well as what are the entries. Some special cases were done by van der Poorten in [1]. He studied solutions to Pell¡¯s equation , ( a nonsquare integer). Using the solution, he derived expansions for the continued fraction of and (with ) and then 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 unfinished paper of Gosper [4]. He develops a set of algorithms for finding closed form expressions of simple functions of a given quadratic irrational, as well as for more complicated functions combining quadratic
%U http://www.hindawi.com/journals/isrn.combinatorics/2013/414623/