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In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…
Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to…
We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…
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…
In the combinatorial theory of continued fractions, the Foata--Zeilberger bijection and its variants have been extensively used to derive various continued fractions enumerating several (sometimes infinitely many) simultaneous statistics on…
We observe that certain famous evaluations of the Rogers-Ramanujan continued fraction $R(q)$ are close to $2\pi-6$ and $(2\pi-6)/2\pi$, and that $2\pi-6$ can be expressed by a Rogers-Ramanujan continued fraction in which $q$ is very nearly…
We give explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
We prove in three ways the basic fact of analysis that complex derivatives are continuous. The first, classical, proof of Cauchy and Goursat uses integration. The second proof of Whyburn and Connell is topological and is based on the…
Euler gives a continued fraction representation of (1+x)^n involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for ``vanishing'' n, and for infinite…
The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the…
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…
In this partly expository paper, we discuss three results. (1) That the two-sided continued fraction of the normalized square root (an important part of the SQUFOF algorithm) has several very attractive properties - periodicity, a symmetry…
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…
Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and…
The direct calculation of the Generalized operator entropy proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient…
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.
Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…
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…
We review Mabry's, Edgar's, and the Viewpoints 2000 Group's proofs without words for the geometric series formula. Mabry and Edgar proved without words that $$\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \cdots\…