Related papers: Continued fractions for complex numbers and values…
We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…
In this paper, we study the complexity of p-adic continued fractions of a rational number, which is the p-adic analogue of the theorem of Lame. We calculate the length of Browkin expansion, and the length of Schneider expansion. Also, some…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…
The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic…
Rational approximations to a square root $\sqrt{k}$ can be produced by iterating the transformation $f(x) = (dx+k)/(x+d)$ starting from $\infty$ for any positive integer $d$. We show that these approximations coincide infinitely often with…
Legendre found that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. Similarly, it is shown here that the continued fraction expansion of $\sqrt…
We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections…
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To…
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…
By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions:…
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…
We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…
Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of…
The connection between continued fractions and orthogonality which is familiar for $J$-fractions and $T$-fractions is extended to what we call $R$-fractions of type I and II. These continued fractions are associated with recurrence…
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…
We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.