Truncation error estimates for Stieltjes fractions

Orthogonal Laurent polynomials and strong moment theory: a survey

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.

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

Analytic theory of continued fractions III : L. Jacobsen (ed.), Lecture Notes in Mathematics 1406. Springer, Berlin, 1989. 142 pp., DM 25, ISBN 3-540-51830-4

Hankel continued fraction and its applications

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

We perform a comprehensive cosmographic analysis of the late-time Universe using the latest Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) baryon acoustic oscillation (BAO) measurements, comparing Taylor, Padé, and Chebyshev expansions as model-independent reconstructions of the background expansion. We consider Padé approximants of order (2,1) and (2,2), a Chebyshev expansion, and a third-order Taylor series. Due to its limited radius of convergence, the Taylor expansion is constrained using only the low-redshift DESI sub-set ($z&amp;lt; 1$), while the rational Padé forms and the Chebyshev expansion are applied over the full DESI DR2 redshift range. Cosmographic parameters are inferred through a Bayesian Markov chain Monte Carlo (MCMC) analysis, and the resulting best-fitting reconstructions of $H(z)$, $d_L(z)$, and BAO distance indicators are compared with the predictions of the Lambda cold dark matter ($\Lambda$CDM) model. All methods are consistent with $\Lambda$CDM at low redshift, but the Chebyshev expansion exhibits noticeable deviations at higher redshifts, while the Padé$_{(2,1)}$ and Padé$_{(2,2)}$ reconstructions remain closely aligned with $\Lambda$CDM across the DESI DR2 range. A model-selection analysis based on Akaike Information Criterion and Bayesian Information Criterion shows a clear statistical preference for the Taylor expansion over low-z$\Lambda$CDM, and a strong preference for Padé cosmography over $\Lambda$CDM when the full DESI DR2 data set is used. These results demonstrate the constraining power of DESI DR2 for cosmographic studies and highlight the utility of rational approximants, especially Padé forms, in extending cosmography reliably to higher redshifts beyond the domain of traditional Taylor series.

Flat Power Series over a Finite Field

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-

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ3 and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

In this paper using a perturbative method a series expansion of the price of a (put up and out) barrier option with time dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and non elementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature and a sufficient condition for the convergence of the perturbation series expansion is given. The numerical experience shows that in the cases of practical interest considered the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options such as step options, step barrier options and so on. The website contains some auxiliary material that helps the understanding of this paper and the computer programs needed to evaluate the formula obtained.

It is well-known that a continued fraction may be regarded as a sequence of Mobius maps. In this partly expository paper we consider continued fractions by examining the action of Mobius maps in hyperbolic space, and in all dimensions. We obtain a general version of the Stern-Stolz theorem valid in all dimensions, and we draw comparisons between the theory of continued fractions and the theories of discrete groups of Mobius maps, and complex dynamics.

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as well as introducing the ideas of a best rational approximation to a real number and exploring some elementary ideas in error analysis. While the paper is a review of some of the classical works on continued factions, it does not present the material in a theorem/proof style. It motivates the results by citing examples and noting some interesting properties that will need to be verified. Notations are given to the references where exact and complete proofs of these facts can be found. The type of investigations done in this paper are motivational and the style of presentation is designed to pique students and teachers curiosity about the subject. Another purpose of this paper is to show that by looking at alternative representations of something that seems to be well known, such as the real numbers, can lead to exciting new and useful outcomes and applications.

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

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].