Convergence Of Continued Fractions Research Articles

On the convergence of periodic continued fractions via linearly recurring sequences

In this paper, we give a new process to study the convergence of periodic continued fractions, making use of the approach on the linear difference equations with periodic coefficients, based on the analytic formula of the linearly recurring sequences. Some new explicit expressions of nth numerators and nth denominators of the convergents of periodic continued fractions lead to discuss their convergence. Moreover, we provide a new discussion on the closed link between quadratic irrational numbers and simple periodic continued fractions. Some applications of the Pell’s equations are given and some numerical examples which illustrate the theory are supplied.

Combinatorial properties of multidimensional continued fractions

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.

Approximation of Differential Operators with Boundary Conditions

The use of spectral methods for solution of boundary value problems is very effective but involves great technical difficulties associated with the implementation of the boundary conditions. There exist several methods of such an implementation, but they are either very cumbersome or require a preliminary analysis of the problem and its reduction to an integral form. We propose a universal means of implementation of the boundary conditions for linear differential operators on a finite interval, which is very simple in its realization. The use of the rational arithmetic allows to assess the effectiveness of this method without interference of the round-off errors. We apply this approach for computation of rational approximations for some fundamental constants. We obtained approximations that in a number of cases are better than those that are given by convergents of regular continued fractions of these constants.

Quantum invariants of hyperbolic knots and extreme values of trigonometric products

In this paper, we study the relation between the function J_{4_1,0}, which arises from a quantum invariant of the figure-eight knot, and Sudler’s trigonometric product. We find J_{4_1,0} up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of J_{4_1,0} to that of Sudler’s trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in response to a question of Lubinsky.

Dilogarithm identities after Bridgeman

AbstractFollowing Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.

Approximation of differential operators with boundary conditions

The use of spectral methods for solution of boundary value problems is very effective but involves great technical difficulties associated with the implementation of the boundary conditions. There exist several methods of such an implementation, but they are either very cumbersome or require a preliminary analysis of the problem and its reduction to an integral form. We propose a universal means of implementation of the boundary conditions for linear differential operators on a finite interval, which is very simple in its realization. The use of the rational arithmetic allows to assess the effectiveness of this method without interference of the round-off errors. We apply this approach for computation of rational approximations for some fundamental constants. We obtained approximations that in a number of cases are better than those that are given by convergents of regular continued fractions of these constants.

Ford Circles Strike Gold

Call a reduced fraction with denominator q a bronze, silver, or gold approximation with respect to a given irrational number ω if they differ by less than the reciprocal of the product of the square of q and 1, 2, or respectively. Suppose A and B are neighboring Farey fractions sandwiching ω, where A’s denominator is at least as large as B’s denominator. Then B is bronze; either A or B is silver; and at least one but not all of A, B, and their mediant is gold. We prove afresh these results using the geometry of Ford circles rather than a purely algebraic approach, and explore some open questions with respect to the bronze-silver-gold classification while panning for gold fractions among the continued fraction convergents for ω.

Dilogarithm identities for solutions to Pell’s equation in terms of continued fraction convergents

We describe a new connection between the dilogarithm function and the solutions of Pell’s equation $$x^2-ny^2 = \pm 1$$ . For each solution x, y to Pell’s equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $$x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]$$ . We further show that Ramanujan’s dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.

Unimodular Invariance of Karyon Expansions of Algebraic Numbers in Multidimensional Continued Fractions

By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.

Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases

Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Haas Molnar Continued Fractions and Metric Diophantine Approximation

Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi’s backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number x, if (p n /q n )n≥1 denotes its sequence of regular continued fraction convergents, set θ n (x) = q 2 |x − p n /q n |, n = 1, 2.... The metric behaviour of the Cesaro averages of the sequence (θ n (x))n≥1 has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence (θ n (x))n≥1 for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of $$({\theta _{{k_n}}}(x))$$ n≥1 for certain sequences (k n )n≥1, initiated by the second named author, to Haas–Molnar maps.

Problems in Calculating Moments and Distribution Functions of Ladder Heights

The problem of approximate calculation of distribution functions of ladder heights is considered in the context of the finite number of its known moments. This problem is solved by means of the Chebyshev continued fractions method. The midpoint is finding the terms of fraction of the nth convergents of continued fractions. The moments are calculated by S. Nagaev’s formulas, including solution of the Frobenius equation and applying the Fa di Bruno formulas for higher derivatives. The problem of high precision calculations is studied.

Research of absolute and figured absolute convergence of the branched continued fractions of the special form

The paper deals with the issue of convergence of one of the generalizations of continued fractions - branched continued fraction (BCF) of the special form with two branches, which has been proposed by the Polish mathematician W. Siemaszko in solving the problem of compliance between the formal double power series and a sequence of rational approximations of the function of two variables. Unlike continued fractions, approximations of which are constructed unambiguously, there are many ways to construct approximations of BCF of the general and special form. The paper examines the conventional approximations of one of the structures of figured approximations of the studied BCF, which is associated with the problem of compliance. The formulas of difference between two approximations (two conventional, two figured, figured and conventional) were given. Using the majorant method and known results of the theory of convergence of continued fractions, the theorem on some sufficient conditions for absolute and absolute figured convergence of BCF of the special form with two branches towards the same border was proved. It was shown that the values of the studied BCF and its approximations belong to a circle, the radius of which depends on the values of BCF elements on the first floor and the constants appearing in the formulation of the theorem. By choosing different values for these constants, it is possible to obtain various signs of absolute and figured absolute convergence of BCF of the special form.

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

Algorithm for summation of divergent continued fractions and some applications

The convergence of continued fractions is defined in a manner other than the conventional definition. A new summation method is used to determine the values of continued fractions and series that diverge in the classical sense. The method is applicable not only to ordinary continued fractions, but also to ones of other classes, for example, to Hessenberg continued fractions. As a result, an original algorithm for finding zeros of nth-degree polynomials is constructed. The r/φ-algorithm proposed is also used to solve infinite systems of linear algebraic equations.

Formulas for rational interpolation and remainders

This paper deals with the existence of interpolating rational functions serving as Thiele continued fraction convergents and also presents the expression for the remainder for such rational interpolations. These problems are similar to multipoint Pade approximations. For the limit case, the expression for the remainder in diagonal Pade approximations at zero is obtained; also a sufficiently simple expression for the exact value of the remainder in the case of the function √1 + z is derived.

On the approximation by three consecutive continued fraction convergents

Denote by pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number x. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all x the pattern θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1, namely 0.12109⋯. All the four other patterns have the same asymptotic frequency 0.18945⋯. The constants are explicitly given.

Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer's theorem that implies the continuous fraction expansions of $e$ and $\tan(1)$, and it can also be derived from Lehmer's work using Fibonacci polynomial identities. We completely characterize those implementations of our limit formula for which the parameter of each Bessel function is the half of an odd integer, allowing them to be replaced with elementary functions.

Periodic representations and rational approximations of square roots

In this paper the properties of Rédei rational functions are used to derive rational approximations for square roots and both Newton and Padé approximations are given as particular cases. As a consequence, such approximations can be derived directly by power matrices. Moreover, Rédei rational functions are introduced as convergents of particular periodic continued fractions and are applied for approximating square roots in the field of p-adic numbers and to study periodic representations. Using the results over the real numbers, we show how to construct periodic continued fractions and approximations of square roots which are simultaneously valid in the real and in the p-adic field.

A formula of difference for one of the figured approximants of two-dimensional continued fractions

The present work deals with one of the methods of constructing the figured approximants (figured convergents) for two-dimensional continued fractions, which is used for studying the conditions of equivalence of two two-dimensional continued fractions. The formula of difference for neighboring approximants, which is established in this work, is applied to investigate the properties of a sequence of such figured convergents of two-dimensional continued fractions.