Asymptotic Expansions for the Coefficients of eP(z)

If P is a real polynomial with non-negative coefficients, thenone can obtain a complete asymptotic expansion for the nth coefficientof e^P(z). The proof requires the following result: the coefficientsin the Stirling expansion for the gamma function are all differentfrom zero. Department of Mathematics and Computer Science Drexel UniversityPhiladelphia PA 19104, USA     

关于Landau常数和Euler-Mascheroni常数的渐近展开式以及Stirling级数的系数

文章给出一个递推关系式来确定Landau常数的渐近展开式的系数.考虑了 Euler-Mascheroni常数和n!的渐近展开式,并给出了递推关系式来确定每个展开式的系数,没有利用Bernoulli数.     

Homogenization and Asymptotic Expansions for Solutions of the Elasticity System with Rapidly Oscillating Periodic Coefficients

Homogenization for a boundary value problem in linear elasticity is discussed. Justification is given for the asymptotic expansion in powers of a small parameter for a solution to the elasticity system in a half-space with rapidly oscillating periodic coefficients.     

Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 111–124, January, 2005.     

Complete Asymptotic Expansions for Altomare Operators

In the present paper, we study an approximation process by a sequence of operators with a prescribed asymptotic behavior. The method of construction of these operators is due to Altomare and Amiar. The operators have several applications in semigroup theory. Our investigation provides an insight in their asymptotic behaviour. Finally, we apply our results to concrete examples of approximation operators recently defined and used by Altomare and Milella for the study of semigroups.     

Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m _n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m _n (nα;β,c) as . One holds uniformly for , and the other holds uniformly for , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases. April 16, 1996. Date revised: October 30, 1996.     

Asymptotic Expansions for Two-Dimensional Hypersingular Integrals

Summary. We define and examine two-dimensional hypersingular integrals on [0,1)² and on [0,)² and relate their Hadamard finite-part (HFP) values to Mellin transforms. These integrands have algebraic singularities of a possibly unintegrable nature on the axes and at the origin. Extending our work on one-dimensional integrals reported in 1998, we obtain variants of the classical Euler-Maclaurin expansion for various two-dimensional integrals. In many cases, the constant term in the expansion (which is not necessarily the leading term) provides the value of the HFP integral. These expansions may be used as the basis for the numerical evaluation of a class of HFP integrals by extrapolation.Mathematics Subject Classification (2000): 65D30, 65B15, 65R10This author was supported in part by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng 38. Part of this work was performed while this author was visiting professor at the Politecnico di Torino, under the sponsorship of the Italian C.N.R.-Gruppo Nazionale per lInformatica Matematica.This author was supported by the Ministero dellUniversitá e della Ricerca Scientifica e Tecnologica of Italy, and by the C.N.R.-Comitato n.11.     

Appell Polynomials and Asymptotic Expansions

Some properties of the Appell polynomials are studied and analyzed. Various formulas and expressions for the Appell quotient are derived and connection with asymptotic expansions is presented.     

Uniform Asymptotic Expansions for Pricing European Options

Starting with a stochastic volatility model, in which the volatility depends on a nonlinear function of a fast varying diffusion, and assuming the fast diffusion is mean reverting, the problem of pricing European options is considered in this paper. Uniform asymptotic expansions of the option price are obtained. The formal expansions are justified and the uniform error bounds are derived using outer and inner expansions of the option prices.     

Complete Asymptotic Expansions for the Chlodovsky Polynomials

The purpose of this article is to study the local rate of convergence of the Chlodovsky operators (C_nf)(x). As the main results, we investigate their asymptotic behaviour and derive the complete asymptotic expansions of these operators. All the coefficients of n^?k (k = 1, 2,…) are calculated in terms of the Stirling numbers of first and second kind. We mention that analogous results for the Bernstein polynomials can be found in Lorentz [2 G. G. Lorentz ( 1953 ). Bernstein Polynomials . University of Toronto Press , Toronto . [Google Scholar]].     

二项式展开的抵消系数以及$q$-摸拟

Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).     

Asymptotic Expansions of Transition Densities for Hybrid Jump-diffusions

A class of hybrid jump diffusions modulated by a Markov chain is considered in this work.Themotivation stems from insurance risk models,and emerging applications in production planning and wirelesscommunications.The models are hybrid in that they involve both continuous dynamics and discrete events.Under suitable conditions,asymptotic expansions of the transition densities for the underlying processes aredeveloped.The formal expansions are validated and the error bounds obtained.     

Exact Remainders for Asymptotic Expansions of Fractional Integrals

This is a continuation of work begun in an earlier paper inwhich we used the theory of distributions to derive explicitexpressions for the remainder terms associated with the asymptoticexpansions of the Stieltjes transform. In this paper similarresults are obtained for the fractional integral of order definedby 1f(x)=1/f()^x_o(x-)^x-1 f(t)dt, >. Heref(t) is a locally integrable function on [0, ) and satisfies f(t) a_st-5–0(ó >0), s=0 as      

Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials t_n(x, N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points x = 0, 1, … , N ? 1, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for t_n(aN, N + 1) in the double scaling limit, namely, N →∞ and n/N → b, where b ∈ (0, 1) and a ∈ (?∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for , and the other involves the Gamma function and holds uniformly for a ∈ (?∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for can be obtained via a symmetry relation of t_n(aN, N + 1) with respect to . Asymptotic formulas for small and large zeros of t_n(x, N + 1) are also given.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

Exact Controllability and Asymptotic Analysis for Shallow Shells

The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2εwith controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickness of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.