A note on extending Euler's connection between continued fractions and power series

where |y1P #1. These were first discussed by Schur [6]1 with |7pl <1 in connection with functions bounded in the unit circle. In ?2 it is shown that to an arbitrary power series there correspond continued fractions (1.1), and, conversely, to every such continued fraction there corresponds a power series which agrees with the power series for the 2pth and (2p+1)th approximants of the continued fraction up to and including the terms involving zP and zP-1, respectively. An important new feature which is given in ?3 is the simple algorithm for the expansion of a power series into a continued fraction (1.1), and, conversely, for the computation of the corresponding power series from the continued fraction. Heretofore this computation has been exceedingly cumbersome. In ?4 conditions under which these continued fractions converge to the corresponding power series are considered, in particular, if the generating function is meromorphic. In ?5 convergence regions for the expansions (1.1) are found, and in ?6 convergence regions for the continued fraction whose approximants are the even and odd approximants of (1.1). In ?7 certain transformations of (1.1) are discussed.

Hankel continued fraction and its applications

Flat Power Series over a Finite Field

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-

In this research article, a new idea concerning the application of concept of the continued fractions to the expansion of the geopotential is treated. The perturbed motion of the satellite in the oblate gravitational field of the Earth using a potential of a continued fractions procedure is studied. To compare with the usual perturbation theory, the continued fraction series is truncated beyond the third term. The first oblateness term of the potential is retained. The equations of motion are constructed using the Lagrange planetary equations. The integrals and averages required for solving the problem is outlined as well as the different cases are highlighted. The perturbations in all orbital elements due to considered model of continued fractions are evaluated. The new findings of the manuscript contrast with classical oblate potential field, the semi-major axis, eccentricity, and inclination, nonzero averaged terms in perturbations are obtained. Extra nonzero averaged perturbing terms in the argument of perigee and ascending node are revealed in addition to the classical perturbations.

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 being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].

Theoretical results are derived for constructing continued fractions which correspond, in some prescribed way, to given power series expansions at two points. In those cases where the two power series are expansions of the same function, new approximations to the functions are obtained. Error estimates for such rational function approximations are developed in certain cases. Some examples are given.

Continued Fractions for Algebraic Formal Power Series over a Finite Base Field

The Berlekamp-Massey algorithm [L. S. De Jong, Numerical aspects of realization algorithms in linearsystems theory, Ph.D. thesis, T. H. Eindhoven, the Netherlands, 1975] and [J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Information Theory, IT-15 (1969), pp. 122–127] for minimal realization problems is a special case of the Padé approximation problem. As a matter of fact, it computes among other polynomials the denominators of the elements of the Padé table that are on the descending diagonal $\{ [0/1],[1/2], \cdots ,[k/k + 1], \cdots \} $ as far as they exist and this algorithm works for nonnormal Padé tables too. This algorithm does not seem to be well-known in Padé approximation literature. It is not very difficult to generalize this algorithm so as to compute the other Padé approximants of a nonnormal table. Some variants will lead to a generalization of the algorithm of Brezinski [Computation of Padé approximants continued fractions, J. Comput. Appl. Math., 2 (1976), pp. 113–123], which computes the descending diagonals of a normal Padé table, and of the algorithm of Watson [D. Bussonnais, “Tous” les algorithmes de calcul par recurrence des approximants de Padé d’une serie, Construction de fractions continues correspondantes, Séminaire d’Analyse Numérique, No. 293, Grenoble,1978], [G. Claessens, new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math., 1 (1975), pp. 141–152], [P. J. S. Watson, Algorithms for differentiation and integration, Padé Approximants and their Applications, P. R. Groves-Morris, ed., Academic Press London, 1973, pp. 93–98] that computes the descending staircases of normal tables. This work is the dual of Cordellier’s [algorithmes de calcul recursif des elements d’une table de Padé non normale, Conference on Padé approximation, Lille, France, 1978], and of McEliece and Shearer [A property of Euclid’s algorithm and an application to Padé approximation, SIAM J. Appl. Math., 34 (1978), pp. 611–616] deriving similar results for ascending diagonals and staircases. A more continued fraction-like approach would generalize the Thatcher algorithm and may be found in [J. A. Murphy and H. R. O’Donohoe, A class of algorithms for obtaining rational approximants to functions which are defined by power series, Z. Angew. Math. Phys., 28 (1977), pp. 1121–1131] and [A. Bultheel, Division algorithms for continued fractions and the Padé table, Applied Mathematics and Programming Division, Rep. TW41, Katholicke Universitiet Leuven, Heverlee, Belgium, August1978].

article Free AccessOn the Computation of Exponential and Hyperbolic Functions Using Continued Fractions Author: Nathaniel Macon Alabama Polytechnic Institute, Auburn, Alabama Alabama Polytechnic Institute, Auburn, AlabamaView Profile Authors Info & Claims Journal of the ACMVolume 2Issue 4Oct. 1955 pp 262–266https://doi.org/10.1145/320809.320813Published:01 October 1955Publication History 3citation497DownloadsMetricsTotal Citations3Total Downloads497Last 12 Months12Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Modeling of municipal waste disposal rates during COVID-19 using separated waste fraction models

Two dimensional hard core bosons suffer strong scattering in the high\ntemperature resistive state at half filling. The dynamical conductivity is\ncalculated using non perturbative tools such as continued fractions, series\nexpansions and exact diagonalization. We find a large temperature range with\nlinearly increasing resistivity and broad dynamical conductivity, signaling a\nbreakdown of Boltzmann-Drude quasiparticle transport theory. At zero\ntemperature, a high frequency peak in the dynamical conductivity appears above\na "Higgs mass" gap, and corresponds to order parameter magnitude fluctuations.\nWe discuss the apparent similarity between conductivity of hard core bosons and\nphenomenological characteristics of cuprates, including the universal scaling\nof Homes et. al. (Nature 430, 539 (2004)).\n

Telescoping continued fractions for the error term in Stirling’s formula

By a result of Hardy-Littlewood concerning the growth of the sums \(\sum\nolimits_{n = 1}^N {{e^{i\pi {n^2}x}}} \), for each s ∈ (1/2, 1], the series \({F_s}(x) = \sum\nolimits_{n = 1}^\infty {{e^{i\pi {n^2}x/{n^s}}}} \) converges almost everywhere but not everywhere on [−1, 1]. However, there does not yet exist an intrinsic description of the set of convergence for Fs. In this paper, we define in terms of even continued fractions a subset of [−1, 1] of full measure where the series converges. As an intermediate step, we prove that for s > 0, the sequence of functions $$\sum\limits_{n = 1}^N {\frac{{{e^{i\pi {n^2}x}}}}{{{n^S}}} - {e^{{\text{sign}}(x)i\frac{\pi }{4}}}|x{|^{S - \frac{1}{2}}}\sum\limits_{n = 1}^{\left\lfloor {N|x|} \right\rfloor } {\frac{{{e^{ - i\pi {n^2}/x}}}}{{{n^S}}}} } $$ converges as N → ∞ to a function Ωs continuous on [−1, 1] {0} with (at most) a singularity at x = 0 of type x(s−1)/2 (s ≠ 1) or a logarithmic singularity (s = 1). We provide an explicit expression for Ωs and the error term. Finally, we study thoroughly the convergence properties of certain series defined in terms of the convergents of the even continued fraction of an irrational number.