Fractions continues normales dans un corps de fonctions hyperelliptiques
Further results on the Euler and Genocchi numbers.
Généralisation de la théorie des fractions continues
Généralisation d'une famille de Shanks
Generalized continued fractions and orbits under the action of Hecke triangle groups
Generalized Lagrange criteria for certain quadratic Diophantine equations.
Halfway to a solution of
It is well known that the continued fraction expansion of readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of . Here we notice that, analogously, the point halfway to a solution of can be recognised. We explain what is going on.
Hausdorff dimension of real numbers with bounded digit averages
Hurwitz and Tasoev continued fractions with long period.
Hurwitz continued fractions with confluent hypergeometric functions
Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions . In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions.
Introduction to Diophantine Approximation
In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].
Involutions Associated With Sums of two Squares
Irrational numbers of constant type -- a new characterization.
Jacobi symbols, ambiguous ideals, and continued fractions
The purpose of this paper is to generalize some seminal results in the literature concerning the interrelationships between Legendre symbols and continued fractions. We introduce the power of ideal theory into the arena. This allows significant improvements over the existing results via the infrastructure of real quadratic fields.
Jednoduché důkazy některých vět o kmenných číslech
Kettenbrüche als Summen ebensolcher
Klein polyhedra and lattices with positive norm minima
A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of . It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the Klein polyhedra generated by a lattice have uniformly bounded determinants...
Lacunary formal power series and the Stern-Brocot sequence
Let be a real lacunary formal power series, where εₙ = 0,1 and . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that is a polynomial if and only if ω ∈ ℤ. In all the other cases is an infinite formal power series; we discuss its algebraic...
Lagrange, central norms, and quadratic Diophantine equations.
Leaping convergents of Hurwitz continued fractions
Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...