Heaps, Continued Fractions, and Orthogonal Polynomials

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

and the above results also follow from the classical theory of continued fractions (for example, see [10] and the remarks in [5]). In view of this, it seems natural to attempt a study of the properties of the polynomials, Pn(x), as determined by the coefficients, cn and Xn. One approach to such an investigation is of course through the study of the convergence of the continued fraction (1.3). In this paper, however, we shall avoid direct reference to the theory of continued fractions although we shall make fairly extensive use of the results from the problem of moments. Instead, our investigation centers upon the true interval of orthogonality of Pn(x) }. [By the true interval of orthogonality, we mean, following Shohat (see [5; 6, p. 113]), the smallest interval which contains in its interior all zeros of all Pn(x).] In this connection, the chain sequences of Wall [10] arise naturally and play a fundamental role. Thus our study is largely arithmetical in character. Throughout this paper, orthogonal means polynomials

Orthogonal Laurent polynomials and strong moment theory: a survey

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

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

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.

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.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

PC-fractions and orthogonal Laurent polynomials for log-normal distributions

Sharp Estimates for Jacobi Matrices and Chain Sequences

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

The ordinary and matrix continued fractions in the theoretical analysis of Hermitian and relaxation operators

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.