Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

Relation between interpolating integral continued fractions and interpolating branched continued fractions

We construct and investigate an interpolating integral continued fraction, which represents a natural generalization of interpolating continued fractions. We also point out the optimal choice of the sequence of interpolating knots.     

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $PSL(2,\mathbb{Z})?\mathbb{H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma ?\mathbb{H}$ and odd and grotesque continued fractions, where $\Gamma ?{\mathbb{Z}}_{3}1{\mathbb{Z}}_3$ is the index two subgroup of $PSL(2,\mathbb{Z})$ generated by the order three elements $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill ?1\hfill & \hfill 1\hfill \end{array}\right)$, and having an ideal quadrilateral as fundamental domain.A similar connection between geodesics on $\Theta ?\mathbb{H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $PSL(2,\mathbb{Z})$ generated by $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

Transcendental continued fractions

In two previous papers Nettler proved the transcendence of the continued fractions $\text{A := a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+ ?}$, $\text{B := b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+ ?}$ as well as the transcendence of the numbers A + B, A ? B, AB, $\text{A}\text{B}$ where the a's and b's are positive integers satisfying a certain mutual growth condition. In the present paper even the algebraic independence of A and B is proved under almost the same condition and furthermore a result concerning the transcendency of A^B is established.     

Quasi-periodic continued fractions

Using recent work of Adamczewski and Bugeaud, we are able to relax the conditions given by Baker to establish transcendence in the class of quasi-periodic continued fractions.     

Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

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.     

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.     

Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to d$d$ real numbers _α1,…,_αd${\alpha }_1,\dots ,{\alpha }_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter t$t$ as t↓0$t\downarrow 0$. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in t$t$.     

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).     

Frequency analysis and continued fractions

The frequency analysis problem is solving by the positive Perron-Carathéodory continued fraction, associated with monic Szegö polynomials.     

Pythagorean approximations and continued fractions

In this article, we will show that the Pythagorean approximationsof coincide with those achieved in the 16th century by means of continued fractions. Assumingthis fact and the known relation that connects the Fibonaccisequence with the golden section, we shall establish a procedureto obtain sequences of rational numbers converging to differentalgebraic irrationals. We will see how approximations to someirrational numbers, using known facts from the history of mathematics,may perhaps help to acquire a better comprehension of the realnumbers and their properties at further mathematics level.     

The Euler-Minding series for branched continued fractions

In the paper “Branched continued fractions for double power series” [J. Comput. Appl. Math. 6 (1980) 121–125] Siemaszko generalizes for branched continued fractions the formula that expresses the difference of two successive convergents of an ordinary continued fraction. However, the generalization is not yet fit to write the branched continued fraction as an Euler-Minding series for the following reason. Indeed a convergent of the branched continued fraction can be written as a partial sum of a series but different convergents are different partial sums of different series. The next convergent cannot be obtained from the previous one by adding some terms. We shall develop here another formula that overcomes this problem.     

The bisection method in higher dimensions

Is the familiar bisection method part of some larger scheme? The aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions. The algorithm preserves the salient features of the bisection method: it is simple, convergence is assured and linear, and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired. Brackets are unions of similar simplexes. An immediate application is to the global minimisation of a Lipschitz continuous function defined on a compact subset of Euclidean space.