Periodic Continued Fractions Research Articles

Statistics of the periods of continued fractions for quadratic irrationals

The distribution of frequencies of elements of continued fractions for random real numbers was obtained by Kuz'min in 1928 and is therefore referred to as Gauss-Kuz'min statistics. An old conjecture of the author states that the elements of periodic continued fractions of quadratic irrationals satisfy the same statistics in the mean. This was recently proved by Bykovsky and his students. In this paper we complement those results by a study of the statistics of the period lengths of continued fractions for quadratic irrationals. In particular, this theory implies that the elements forming the periods of continued fractions of the roots of the equations with integer coefficients do not exhaust the set of all random sequences whose elements satisfy the Gauss-Kuz'min statistics. For example, these sequences are palindromic, that is, they read the same backwards as forwards.

Representation of algebraic numbers by periodic branching continued fractions

The notion of a periodic branching continued fraction is a natural generalization of the notion of a periodic continued fraction. It is shown that for an arbitrary positive algebraic number one can construct a periodic branching continued fraction with natural elements convergent to this number.

Shrinking the period lengths of continued fractions while still capturing convergents

Here we prove that every real quadratic irrational α can be expressed as a periodic non-simple continued fraction having period length one. Moreover, we show that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of α. We close with an application relating the structure of a quadratic α to its conjugate.

Some Continued Fractions in Ramanujan’s Lost Notebook

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

A continued fraction from Ramanujan’s lost notebook

In his lost notebook, Ramanujan recorded without a proof the value of an elegant periodic continued fraction. The authors provide an elementary proof of a corrected version of Ramanujan’s claim.

Periodic continued fraction representations of different quark’s mass ratios

In the present work, first we give some theorems on continued fraction expansions. As shown in [Chaos, Solitons & Fractals 21 (2004) 9–19] the limit set of the Kleinian transformation acting on the E-infinity Cantorian space-time is a set of periodic continued fractions. We expand all possible quark’s mass ratios as periodic continued fractions and following [Chaos, Solitons & Fractals 22 (2004) 749–752], we establish that their vibrations are like fractal polylines.

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E -infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.

Some results concerning certain periodic continued fractions

Some results concerning certain periodic continued fractions

Episturmian morphisms and a Galois theorem on continued fractions

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w . Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

The unreasonable effectualness of continued function expansions

AbstractMany generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E -infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.

On a connection between the limit set of the Möbius–Klein transformation, periodic continued fractions, El Naschie’s topological theory of high energy particle physics and the possibility of a new axion-like particle

In the present work we first give a general representation of the derivatives of the irrational number φ, for instance 1 φ , 1 φ 2 , 1 φ 3 etc., as periodic continued fractions. Any irrational number can then be expanded in an infinite continued fraction. The limit set of the Kleinian transformation acting on the E-infinity Cantorian spacetime turned out to be this set of periodic continued fractions, consequently the vacuum of the E-infinity is described by this limit set. As discussed by El Naschie, every particle can be interpreted geometrically as a scaling of another. This is done using the topology of hyperbolic Kleinian space of VAK, which is nothing but our limit set. Here we will present the ratios of the theoretical masses of certain elementary particles to that of some chosen particles in term of φ. Many of these masses are quite close to integer multiples of the mass of a chosen particle. Finally we discuss the possibility of new transfinite, axion-like particles as discussed recently by Krauss and El Naschie [Quintessence, Vintage, London, 1999].

Renormalizations and Rigidity Theory for Circle Homeomorphisms with Singularities of the Break Type

Circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion. We establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result. Namely, we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same ``size'' of a break.

Quasi-Elliptic Integrals and Periodic Continued Fractions

In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral $$$$

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of p-adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every m, 1 < m < 5000, 5 m if ∨m E Q 5 Q, then ∨m has a periodic continued fraction expansion. The same is not true in Q p for some larger values of p.

Improved truncation error bounds for limit periodic continued fractions with additional assumptions on its elements

It is well known that for convergent pure periodic continued fractions (i.e.: | r|=| x 1/ x 2|<1) the truncation error is O(| r| n ). In earlier articles, with assumptions weaker than ∑ m=1 ∞ md m<∞ , it was shown that the truncation error for limit periodic continued fractions is, at best, of the form K(|r′|)|r′| n, 0<|r′|<|r| , where K(| r′|) is a function of | r′| which may tend to infinity as | r′|→| r|. It is thus of interest to determine whether there exist conditions on {a n}, {b n} in the limit periodic continued fraction K( a n / b n ), which would insure that the truncation error is O(| r| n ). It is shown here that restrictions of that kind do exist and that ∑ nd n <∞ is such a condition. In view of the result on pure periodic continued fractions, mentioned above, the estimate here obtained is optimal. Whether, beyond its aesthetic appeal, this optimal error bound might also be useful, is not known to the author.

On the convergence of periodic integral continued fractions with variable limits of integration

We establish estimates for the “tails” of periodic integral continued fractions with variable upper limits of integration. We prove a theorem on the uniform and absolute convergence of such fractions, and we obtain an estimate of their rate of convergence.

Model-Building Problem of Periodically Correlatedm-Variate Moving Average Processes

The model-building problem of periodically correlatedm-variateq-dependent processes is considered. We show that for a given periodical autocovariance function of anm-variateMA(q) process there are two particular corresponding classes (that may reduce to one class) of periodic (equivalent) models. Furthermore, any other (intermediate) model is not periodic. It is, however, asymptotically periodic. The matrix coefficients of the particular periodic models are given in terms of limits of some periodic matrix continued fractions, which are a generalization of the classical periodic continued fractions (Wall, 1948). These periodic matrix continued fractions are particular solutions of some prospective and/or retrospective recursion equations, arising from the symbolic factorization of the associated linear autocovariance operator. In addition, we establish a procedure to calculate these limits. Numerical examples are given for the simple cases of periodically correlated univariate one- and two-dependent processes.

Algebraic interpretation of continued fractions

An alternative (equivalent) definition of continued fractions in terms of a group representation is introduced. With this definition, continued fractions are considered as sequences in a topological group, converging (in some sense) to its boundary. This point of view yields an alternative (equivalent) proof for Lane's convergence theorem for periodic continued fractions.