Asymptotic summation of power series

In this paper is examined the acceleration of convergence of alternative and non-alternative numerical and power series by means of an Euler-Abel type operator, that is defined earlier by G.A.Sorokin and I.Z.Milovanovic. Through this type of linear operator of generalized difference of sequence is achieved that alternative and non-alternative numerical and power series to be transformed into series with higher speed of convergence than the initial series. At the end of this paper is given the implementation of this method through a numeric example.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

Summation of power series by self-similar factor approximants

We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate J v ( x ) ( | x | small ) {J_v}(x)\;(|x|\;{\text {small}}) , e x K v ( x ) ( Re ⁡ x > 0 , | x | large ) {e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) , and the modulus and phase of H v ( 1 ) ( x ) ( Re ⁡ x > 0 , | x | large ) H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) . The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for K v + 1 ( x ) / K v ( x ) {K_{v + 1}}(x)/{K_v}(x) the convergence of which increases with | x | |x| . The procedures were tested numerically both for integer and fractional values of v , and for real and complex x .

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.

For a Dirac particle, the general mathematical study of the particle tunneling process through an effective potential barrier generated by the Schwarzschild black hole background is done. The study is based on the use of 8 Frobenius’ solutions of the related second-order differential equation with 3 regular and 2 irregular singularities of the rank 2. Solutions of the radial equations are constructed in explicit form, and the convergence of the involved power series is proved in the physical range f the variable (1, ). r∈ +∞ Results for the tunneling effect are significantly different for two situations: one when the particle falls on the barrier from the inside and another when the particle falls from the outside. The mathematical structure of the derived asymptotic relations is exact, however the analytical expressions for the involved convergent powers series are unknown, and a further study of penetration and reflection coefficients should be based on the numerical summation of the power series.

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

We consider t-ary trees characterized by their numbers of nodes and their total path length. When t=2 these are called binary trees, and in such trees a parent node may have up to t child nodes. We give asymptotic expansions for the total number of trees with nodes and path length p, when n and p are large. We consider several different ranges of n and p. For n→∞ and p=O(n^{3/2}) we recover the Airy distribution for the path length in trees with many nodes, and also obtain higher order asymptotic results. For p→∞ and an appropriate range of n we obtain a limiting Gaussian distribution for the number of nodes in trees with large path lengths. The mean and variance are expressed in terms of the maximal root of the Airy function. Singular perturbation methods, such as asymptotic matching and WKB type expansions, are used throughout, and they are combined with more standard methods of analytic combinatorics, such as generating functions, singularity analysis, saddle point method, etc. The results are applicable to problems in information theory, that involve data compression schemes which parse long sequence into shorter phrases. Numerical studies show the accuracy of the various asymptotic approximations. Key Words: Trees; Universal Types; Asymptotics; Path Length; Singular Perturbations

From the exact solution of an elliptic boundary value problem in a sector, asymptotic approximations with respect to a small parameter are derived. The asymptotic expansion is uniformly valid in the boundary layers. Also the phenomena for the case of almost characteristic boundaries are discussed.

Let \(K_{n}^{N}(x;p,q)\) be the Krawtchouk polynomials and μ = N/n. An asymptotic expansion is derived for \(K_{n}^{N}(x;p,q)\), when x is a fixed number. This expansion holds uniformly for μ in [1,∞), and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of \(K_{n}^{N}(x;p,q)\) in various cases depending on the values of p, q and μ.

A closed processor-sharing queueing model that corresponds to M terminals in series with a central processing unit (CPU) is considered under heavy usage conditions. The effects of switching times between the CPU and the individual jobs are included by assuming that the effective service rate that the CPU provides is a decreasing function of the instantaneous number of jobs being processed. Using perturbation methods, asymptotic expansions are constructed for the first two moments of a customer's response (sojourn) time, conditioned on that customer's total required service from the CPU. The asymptotic limit assumes that the number of terminals M is large $( M \gg 1 )$, that the rate at which the CPU provides service is fast $( O( M ) )$, and that the duration of a typical switching time is small compared to the time the CPU actually spends servicing the individual jobs. Extensive numerical comparisons show the quality of our asymptotic approximations.

An asymptotic approximation is developed for evaluating the probability integrals that arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on second-order reliability methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte Carlo simulation. Simple structures subjected to white noise stochastic excitation are used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method.

The bibliography of articles that can be classed as applications of Borel’s methods has about 70 entries. Of these, only one predates 1970. That is the article by Cunningham in 1905 A[18] entitled An extension of Borel’s exponential method of summation of divergent series applied to linear differential equations. Here, Cunningham’s object was to make more precise the connection between Laplace’s integral solutions of linear differential equations with rational coefficients and of rank unity in the neighbourhood of z = ∞, and the power series solution with an exponential factor. He noted that Poincaré had developed asymptotic expansions of the integrals, and had found them useful for numerical calculations. However, these asymptotic expansions were of little value in that they did not represent unique analytic functions.