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.     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.     

An algorithm for constructing multidimensional continued fractions and linear dependence of numbers

The Güting algorithm for constructing multidimensional continued fractions is considered. It is proved that, in the case of dimension 2, this algorithm can be used to find the coefficients of the linear dependence of numbers; a criterion is given for verifying that the partial quotients furnished by the algorithmare, indeed, elements of the continued fraction for the expanded (generally irrational) numbers.     

Convergence acceleration of continued fractions of Poincaré's type 1

A new method of convergence acceleration is proposed for continued fractions of Poincaré's type 1. Each step of the method (and not only the first one, as in the Hautot method [1]) is based on an asymptotic behaviour of continued fraction tails. A theorem is proved detailing properties of the method in six cases considered here. Results of numerical tests for all Hautot's examples confirm a good performance of the method.     

基于二元连分式的散乱数据递推插值格式

本文首先基于新的非张量积型偏逆差商递推算法,分别构造奇数与偶数个插值节点上的二元连分式散乱数据插值格式,进而得到被插函数与二元连分式间的恒等式.接着,利用连分式三项递推关系式,提出特征定理来研究插值连分式的分子分母次数.然后,数值算例表明新的递推格式可行有效,同时,通过比较二元Thiele型插值连分式的分子分母次数,发现新的二元插值连分式的分子分母次数较低,这主要归功于节省了冗余的插值节点. 最后,计算此有理函数插值所需要的四则运算次数少于计算径向基函数插值.     

基于Samelson逆的向量值连分式的收敛准则

The aim of this work is to give some criteria on the convergence of vector valued continued fractions defined by Samelson inverse. We give a new approach to prove the convergence theory of continued fractions. First, by means of the modified classical backward recurrence relation, we obtain a formula between the m-th and n-th convergence of vector valued continued fractions. Second, using this formula, we give necessary and sufficient conditions for the convergence of vector valued continued fractions.     

Analysis of non-linear dynamic systems excited by intense non-white noise

A method of finding the stationary moments of the solution of non-linear stochastic equations with additive Gaussian random action is proposed, based on the use of matrix continued fractions. The method imposes no a priori limitations on the intensity and correlation time of the noise. Two methods of constructing such fractions are considered, namely, based on a chain of equations for the combined moments or a chain of equations for the combined cumulants of the solution and a random force.     

A note on extending Euler's connection between continued fractions and power series

Euler's Connection describes an exact equivalence between certain continued fractions and power series. If the partial numerators and denominators of the continued fractions are perturbed slightly, the continued fractions equal power series plus easily computed error terms. These continued fractions may be integrated by the series with another easily computed error term.     

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.     

The origins of Euler's early work on continued fractions

In this paper, I examine Euler's early work on the elementary properties of continued fractions in the 1730s, and investigate its possible links to previous writings on continued fractions by authors such as William Brouncker. By analysing the content of Euler's first paper on continued fractions, ‘De fractionibus continuis dissertatio’ (1737, published 1744) I conclude that, contrary to what one might expect, Euler's work on continued fractions initially arose not from earlier writings on continued fractions, but from a wish to solve the Riccati differential equation.     

Convergence of matrix continued fractions

The aim of this work is to give some criteria on the convergence of matrix continued fractions. We begin by presenting some new results which generalize the links between the convergent elements of real continued fractions. Secondly, we give necessary and sufficient conditions for the convergence of continued fractions of matrix arguments. This paper will be completed by illustrating the theoretical results with some examples.     

The St. Petersburg game and continued fractions

The St. Petersburg game is a well known example of a sequence of i.i.d. random variables with infinite expectation and was considered as a paradox since no single “fair” entry fee exists. This Note shows how the sequence of continued fraction digits of a random real number makes a reasonable choice of entry fees. Moreover, known results for continued fractions can be obtained for the St. Petersburg game using exactly the same proofs and these results explain exactly how the player is favoured even with a fair entry fee (thus resolving a point made by Aaronson).     

Evaluation of Branched Continued Fractions using Block-Tridiagonal Linear Systems

The convergent of an ordinary continued fraction can be computedby solving a tridiagonal linear system for its first unknown.In this paper, this approach is generalized to branched continuedfractions, and it is shown how the convergent of a branchedcontinued fraction can be considered as the first unknown ofa block-tridiagonal linear system. Hence algorithms for thesolution of such systems of equations can be used for the computationof convergents of branched continued fractions, which have applicationsin approximation theory, systems theory, etc. In future research,special attention will be paid to the use of parallel algorithms.     

RESEARCH ANNOUNCEMENTS Continued Fractions of Functions Connec …

The theory of continued fractions is very useful in studying real quadratic number fields (see ［2-5］).E. Artin in ［1］ introduced continued fractions of functions to study quadratic function fields, using formal Laurent expansions, which isessentially the theory of completion of the function fields at the infinite valuation. Here we first re-developthe theory of continued fractions of functions in a more elementary and manipulable manner mainly using long division of polynomials; and then study properties of the continued fractions, which will have important applications in studying quadratic function fields obtaining remarkable results on unit groups, class groups, and class numbers.     

Continued fractions and the second Kepler law

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.     

A theorem of Joseph-Alfred Serret and its relation to perfect quantum state transfer

In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.     

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

Solution formulas for linear difference equations with applications to continued fractions

For any system of linear difference equations of arbitrary order, a family of solution formulas is constructed explicitly by way of relating the given system to simpler neighboring systems. These formulas are then used to investigate the asymptotic behavior of the solutions. When applying this difference equation method to second-order equations that belong to neighboring continued fractions, new results concerning convergence of continued fractions as well as meromorphic extension of analytic continued fractions beyond their convergence region are provided. This is demonstrated for analytic continued fractions whose elements tend to infinity. Finally, a recent result on the existence of limits of solutions to real difference equations having infinite order is extended to complex equations.