Chain sequences and orthogonal polynomials

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

Sharp Estimates for Jacobi Matrices and Chain Sequences

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two‐level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.

Orthogonal Laurent polynomials and strong moment theory: a survey

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

Perhaps the most significant result of this work is the following uniform bound on the orthogonal polynomials throughout the interval of orthogonality:

We study the spectrum of a parameter-dependent Sturm-Liouville problem. By using, as the main tool, the theory of continued fractions we obtain a characterization of the eigenvalues. From here estimates for large eigenvalues, depending on the parameter, and an asymptotic result for the lowest eigenvalue will follow. We associate to each given eigenvalue two sequences converging monotonically to the eigenvalue itself, one from above and the other from below. To obtain this result we use the theory of orthogonal polynomials.

These notes are an exposition of work of Francon–Viennot, Flajolet and Viennot dealing with combinatorial aspects of the theory of continued fractions. We have added here some of the missing details, some of the proofs that were not given in the original works, and hopefully have corrected some of the errors without creating new ones.

The Early History of the Moment Problem

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.

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

A connection between the symbolic description of the geodesic flows on certain modular surfaces and the theory of continued fractions is developed. The well-known properties of these dynamical systems then lead to some new results about continued fractions.

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

The quotient-difference (=QD) algorithm developed by the author may be considered as an extension ofBernoulli's method for solving algebraic equations. WhereasBernoulli's method gives the dominant root as the limit of a sequence of quotientsq 1 () =s 1 (+1) /s 1 () formed from a certain numerical sequences 1 () , the QD-algorithm gives (under certain conditions) all the rootsλ σ as the limits of similiar quotient sequencesq σ () =s σ (+1) /s σ () . Close relationship exists between this method and the theory of continued fractions. In fact the QD-algorithm permits developing a function given in the form of a power series into a continued fraction in a remarkably simple manner. In this paper only the theoretical aspects of the method are discussed. Practical applications will be discussed later.