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.     

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]].     

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.     

Krawtchouk多项式零点的渐近展开及其误差限

Krawtchouk多项式在现代物理学中有着广泛应用．基于Li和Wong的结果,利用Airy函数改进了Krawtchouk多项式的渐近展开式,而且得到了一个一致有效的渐近展开式A·D2进一步,利用Airy函数零点的性质,推导出了Krawtchouk多项式零点的渐近展开式,并讨论了其相应的误差限．同时还给出了Krawtchouk多项式和其零点的渐近性态,它优于Li和Wong的结果．     

Combinatorial Expansions of Kazhdan-Lusztig Polynomials

We introduce two related families of polynomials, easily computableby simple recursions into which any Kazhdan–Lusztig (andinverse Kazhdan–Lusztig) polynomial of any Coxeter groupcan be expanded linearly, and we give combinatorial interpretationsto the coefficients in these expansions. This yields a combinatorialrule for computing the Kazhdan–Lusztig polynomials interms of paths in a directed graph, and a completely combinatorialreformulation of the nonnegativity conjecture [15, p. 166].     

Baskakov-Durrmeyer type operators involving generalized Appell Polynomials

In this paper, we define the Baskakov-Durrmeyer type operators based on generalized Appell polynomials. Here, we establish moment estimates, an estimate via weighted modulus of continuity and a Voronovoskaya type asymptotic result. Further, we study a quantitative-Voronovoskaya-type theorem and Grüss Voronovskaya-type theorem. Lastly, we give the approximation result for functions having derivatives of bounded variation.     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Convergence of Sums of Appell Polynomials with Infinite Variance

We investigate the convergence of distributions of partial sums of Appell polynomials of a long-memory moving average process X _t with i.i.d. innovations _s in the case where the variance , and the distribution of #x03BE; ₀ ^m belongs to the domain of attraction of an -stable law with 1<< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an -stable Lévy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of , m, and d, where 0X _t.     

Expansions in Terms of q-Laguerre Heat Polynomials

In a previous paper M. S. Ben Hammouda and Akram Nemri derived criteria for the expansion of solutions u(x, t) from the generalized q-heat equation, in series of polynomial solutions \({p_{n}^{\alpha}}\) , thus extending an analogous theory of the ordinary heat equation developed by P. C. Rosenbloom and D. V. Widder. It is the goal to carry out a parallel study for the q-Laguerre differential heat equation and establish the region of convergence of the series of q-Laguerre heat polynomials and their temperature transforms.     

Expansions for the Fundamental Hermite Interpolation Polynomials in Terms of Chebyshev Polynomials

We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial.     

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.     

Biased Expansions of Biased Graphs and Their Chromatic Polynomials

A biased graph is a graph with a distinguished set of circles, such that if two circles in the set are contained in a theta graph, then so is the third circle of the theta graph. We introduce a new biased graph, a biased expansion of a biased graph, that satisfies certain lifting and projection properties with the original biased graph. We relate the chromatic polynomials of a biased graph and its biased expansions, thus generalizing a biased-graph result of Zaslavsky [7] and a hyperplane result of Ehrenborg and Readdy [1]. We also determine which biased expansions have supersolvable bias matroids.     

Asymptotic Expansions at Nonsymmetric Cuspidal Points

Mathematical Notes - We study the asymptotics of solutions to the Dirichlet problem in a domain $$\mathcal{X} \subset \mathbb{R}^3$$ whose boundary contains a singular point $$O$$ . In a small...     

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.     

Asymptotic Representation of Zolotarev Polynomials

In 1868 Zolotarev determined the polynomial which deviates leastfrom zero with respect to the maximum norm on [–1,1] amongall polynomials of the form where R is given. The polynomial was given explicitly in termsof elliptic functions by Zolotarev. It is now called the Zolotarevpolynomial. Zolotarev also gave an explicit expression for theminimum deviation. In the sequel attempts have been made toreplace the elliptic functions and to express the Zolotarevpolynomial and the minimum deviation in terms of elementaryfunctions, at least asymptotically. In 1913 Bernstein succeededin finding an asymptotic formula for the minimum deviation,which has been improved several times since then. Here we givethe first asymptotic representation of the Zolotarev polynomials.For the asymptotic representation we use the rational functionsintroduced by Bernstein. Furthermore, we obtain asymptotic representationsof minimal polynomials with interpolation constraints whichare of interest in the theory of the iterative solution of inconsistentlinear systems of equations.     

Krawtchouk多项式的渐近性态

在前人的基础上,对Krawtchouk多项式及其零点的渐近性态进行了研究.首先推导出对于任意固定的u=n/N∈(0,P)或(0,q)Krawtchouk多项式Kn(λN)(其中λ=xN,0<λ<1)的一致有效渐近展开式.然后又得到了它的零点的渐近性态,并对其相应的误差限进行分析.该误差限为o(n-4/3).     

Infinite Families of Pellian Polynomials and Their Continued Fraction Expansions

We investigate families $ \lbrace D_k(X)\rbrace_{k\in{\rm N}} $ of quadratic integral polynomials and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is constant. Furthermore, we show that the period lengths of $ \sqrt {D_k(X)} $ go to infinity with k. For each member of the families involved, we show how to easily determine the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is related to that of $ \sqrt {C} $ . This continues work in [3]-[5].     

Linear Processes, Long-Range Dependence and Asymptotic Expansions

Linear processes have a wide range of applications in time series analysis. This paper reviews some recent results by the author on the limit theory for the functionals of linear processes.