Family Of Continued Fractions Research Articles

Convergence conditions for p-adic continued fractions

Continued fractions have been introduced in the field of p-adic numbers Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. In particular, an analogue of the Lagrange’s Theorem is not yet proved for any attempt of generalizing continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. Thus, it is worth to study the definition of new algorithms for p-adic continued fractions. The main condition that a new method needs to fulfill is the convergence in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} of the continued fractions. In this paper we study some convergence conditions for continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.

Continued fractions arising from $${\mathcal F}_{1,3}$$ F 1 , 3

We study a family of continued fractions arising from a graph known as $${\mathcal F}_{1,3}$$ which is isomorphic to a subgraph of the Farey graph. We call these continued fractions $${\mathcal F}_{1,3}$$ -continued fractions. In fact, certain paths from infinity to a vertex in $${\mathcal F}_{1,3}$$ correspond to finite $${\mathcal F}_{1,3}$$ -continued fractions of the vertex and vice versa. Further, we study uniqueness of the longest $${\mathcal F}_{1,3}$$ -continued fraction expansions of real numbers and show that their convergents are best approximations of the numbers by vertices of $${\mathcal F}_{1,3}$$ .

A family of continued fractions

We investigate a one-parameter family of infinite generalised continued fractions. The fractions converge to rational values which can be explicitly evaluated. The sequences of numerators and denominators are already to be found in the Online Encyclopedia of Integer Sequences with reference to various topics – but not to continued fractions.

Computation of special functions by Padé approximants with orthogonal polynomial denominators

This paper deals with the computation of special functions of mathematics and physics in the complex domain using continued fraction (one-point or two-point Pade) approximants. We consider three families of continued fractions (Stieltjes fractions, real J-fractions and non-negative T-fractions) whose denominators are orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical roundoff and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Results from computational experiments are given to illustrate the methods.

Sur le développement d'une fraction continue liée à la série hypergéométrique et son interprétation en termes de records et anti-records dans les permutations

A particular family of continued fractions, derived from the hypergeometric series, gives rise to two sets of polynomials Fn,(a, b) and Cn(a, b). The coefficient of a'bs in Fn(a, b) enumerates the permutations of {1, 2, . . . , n} with r ‘records’ and s ‘exclusive anti-records’. If only ‘connected’ permutations are taken into account, all the anti-records are exclusive; the relevant coefficients are those of a'bs in Cn (a, b) and depend symmetrically on r and s.