New backward recurrences for Bessel functions

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.

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.

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.

Correct short time propagator for Feynman path integration by power series expansion in Δ t

: The usual power series representations for hypergeometric functions in two variables have a limited range of validity. In particular, they are of little use when the magnitude of one of the variables becomes very large. Using a Barnes-type integral representation and the concept of analytic continuation, the region of utility is extended to the desired domain. The poles that occur in the Barnes-type integral are assumed to be simple. Thus, explicit asymptotic expansions are obtained for each of the fourteen hypergeometric functions that belong to this class. Hypergeometric functions, Functions with a large argument, Functions of two variables, Contour integration, Power series expansions.

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

We consider the asymptotic behavior of solutions of the first -difference Painlevé equation in the limits and . Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behavior associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behavior is described by a power series expression, and find that the Stokes curves are described by curves known as -spirals.

23 - Unified Treatment of Ideal Fermi, Bose, and Classical Gases

In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given.

De Moivre and Bell polynomials

We give an asymptotic expansion in powers of $n^{-1}$ of the remainder $\sum_{j=n}^\infty f_jz^j$ , when the sequence $f_n$ has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series ( $z=1$ ), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of $S(z;j_0,\nu,a,b,p)=\sum_{j=j_0}^\infty z^j$ $(j+b)^{\nu-1}(j+a)^{-p}$ , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series.

We present a novel and completely deterministic method to model chaotic orbits in nonlinear discrete dynamics, taking the quadratic map as an example. This method is based on the resurgent analysis developed by Écalle to perform the resummation of divergent power series given by asymptotic expansions in linear differential equations with variable coefficients. To determine the long-term behavior of the dynamics, we calculate the zeros of a function representing the unstable manifold of the system using Newton’s method. The asymptotic expansion of the function is expressed as a kind of negative power series, which enables the computation with high accuracy. By use of the obtained zeros, we visualize the set of homoclinic points. This set corresponds to the Julia set in one-dimensional complex dynamical systems. The presented method is easily extendable to two-dimensional nonlinear dynamical systems such as Hénon maps.

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series∑nQnzn, whereQn, which depends on the temperatureT, is the “partition function” of a cluster of sizen. At low temperatures this series has a finite radius of convergencezs. Some theorems are proved showing that ifQn, considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromzs to +∞ and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Doring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.

This paper presents a method of generating the reasonable approximations to aerodynamic transfer functions on general configurations of elastic wings or wing-body combinations. The transfer functions are specified by asymptotic expansions in which either the polynomial or rational function approximation is applied. Procedures are described for incorporating a variety of quantitative and qualitative information concerning the unsteady aerodynamic behavior in both subsonic and supersonic flow regimes. Two types of power series approximations, power series with real coefficients and the other with complex coefficients, are applied. Consideration of the hereditary effect of wake vortices on the wing-bound vortices and the analogy with the behavior of a feedback control system provide useful guidance in selecting the form of approximation in both subsonic and supersonic flows. By relaxing the restriction imposed in the first-order piston theory, a limit as co^oo is discussed. Both the standard and extended Fade approximates s = # and s = R are studied in order to improve their accuracy at high k values. The above procedures are programmed in conjunction with the lifting surface theory, doublet lattice, and advanced Mach box scheme and demonstrated for the determination of AGARD wing aerodynamics and the flutter stability boundary for advanced aircraft configurations. The results appear to be satisfactory.

Methods for the computation of real parabolic cylinder functions U ( a , x ), and V ( a , x ) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying article we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.