Related papers: Simple Continued Fractions an Approach for High Sc…
The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an…
Recently, W. M. Schmidt and L. Summerer developed a new theory called Parametric Geometry of Numbers which approximates the behaviour of the successive minima of a family of convex bodies in $\mathbb{R}^{n}$ related to the problem of…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…
This paper is one of a series in which elementary-education practice is analyzed by comparison with the history of mathematics, mathematical structure, modern practice, and (occasionally) cognitive neuroscience. The primary concerns are:…
There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose…
$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…
The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function…
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future…
We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…
There are infinite processes (matrix products, continued fractions, $(r,s)$-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a…
The algorithm of Shor for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing…
In this paper we define a new type of continued fraction expansion for a real number $x \in I_m:=[0,m-1], m\in N_+, m\geq 2$: \[x = \frac{m^{-b_1(x)}}{\displaystyle 1+\frac{m^{-b_2(x)}}{1+\ddots}}:=[b_1(x), b_2(x), ...]_m. \] Then, we…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
We give a concise introduction to the theory of continuants and show how Perron used them in his proof of Tietze theorem on the convergence of infinite semi-regular continued fractions, as well as for the study of the convergence of purely…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition…