Related papers: Simple Continued Fractions an Approach for High Sc…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
This paper continues the author's previous studies on continued fractions and Heron's algorithm, as from his former JMM2017 presentation (see \cite{CF.HA}).\par\medskip Extending the notion of continued fraction to the $p$-adic fields, one…
It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…
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 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…
The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify…
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.
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued…
Perfect fractals are mathematical objects that, because they are generated by recursive processes, have self-similarity and infinite complexity. In particular, they also have a fractional dimension. Although several proposals for the study…
We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than…
A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge…
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.