Related papers: A multidimensional continued fraction generalizati…
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.
Stern's diatomic series, denoted by $(a(n))_{n \geq 0}$, is defined by the recurrence relations $a(2n) = a(n)$ and $a(2n + 1) = a(n) + a(n + 1)$ for $n \geq 1$, and initial values $a(0) = 0$ and $a(1) = 1$. A record-setter for a sequence…
The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…
In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T_slow and T_fast: these maps are defined for (x,y) in [0,1], where x is the number…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…
The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the…
We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b>0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and…
Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…
Triangle partition maps form a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This paper begins the exploration of the functional analysis behind the transfer operator of each of these…
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions,…
Analytical expressions are derived for the position of irreducible fractions in the Farey sequence $F_N$ of order $N$ for a particular choice of $N$. The asymptotic behaviour is derived obtaining a lower error bound than in previous results…
Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$…
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…
In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.
The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary…
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…
This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also…
We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is…