ON ABEL-GONTSCHAROFF-GOULD’S POLYNOMIALS

Abstract In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomialsis presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actu-ally equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of relatedexpansion formulas involving Gontscharoff's remainder and a new form of it are demonstrated, and also il-lustrated with several examples.     

ON COEFFICIENT POLYNOMIALS OF CUBIC HERMITE-PAD(?) APPROXIMATIONS TO THE EXPONENTIAL FUNCTION

The polynomials related with cubic Hermite-Padéapproximation to the exponentialfunction are investigated which have degrees at most n,m,s respectively.A connectionis given between the coefficients of each of the polynomials and certain hypergeometricfunctions,which leads to a simple expression for a polynomial in a special case.Contourintegral representations of the polynomials are given.By using of the saddle point methodthe exact asymptotics of the polynomials are derived as n,m,s tend to infinity throughcertain ray sequence.Some further uniform asymptotic aspects of the polynomials are alsodiscussed.     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

TRACING A PLANAR ALGEBRAIC CURVE

In this paper, an algorithm that determines a real algebraic curve is outlined. Its basicstep is to divide the plane into subdomain1s that include only simple branches of the algebraic curvewithout singular points. Each of the branches is then stably and efficiently traced in the particularsubdomain. Except for tracing, the algorithm requires only a couple of simple operations on poly-nomials that ran be carried out exacrly if the coefficients are rational, and the determination of the real roots of several univariate polynomials.     

ASYMPTOTIC BEHAVIOR OF ECKHOFF＇S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.     

STANCU POLYNOMIALS BASED ON THE Q-INTEGERS

A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper.Moreover,the shape-preserving and convergence properties of these polynomials are also investigated.     

On an extended Hardy-Hilbert‘s inequality in the whole plane

By means of weight coefficients, a complex integral formula and Hermite-Hadamard’s inequality, a new extended Hardy-Hilbert‘s inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions and a few particular cases are considered.     

纽结与多项式

In this paper, we deal with some corresponding relations between knots and polynomials by using the basic properties of knot polynomials （such as, some special values of knot polynomials, the Arf invariant and derivative of knot polynomials）. We give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than five, is the Jones polynomial of a certain knot.     

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved.Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given.Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced.As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems,two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.     

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.     

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45     

Equidistribution of points via energy

We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients.     

On the combinatorics of the Boros-Moll polynomials

The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan’s Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata’s bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.     

An <Emphasis Type="Italic">ABC</Emphasis> inequality for Mahler’s measure

We prove an upper bound for the Mahler measure of the Wronskian of a collection of N linearly independent polynomials with complex coefficients. If the coefficients of the polynomials are algebraic numbers we obtain an inequality for the absolute Weil heights of the roots of the polynomials. This later inequality is analogous to the abc inequality for polynomials, and also has applications to Diophantine problems. Research supported in part by the National Science Foundation (DMS-06-03282) and the Erwin Schr?dinger Institute. Author’s address: Department of Mathematics, University of Texas, Austin, Texas 78712, USA     

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.     

Extraneous fixed points of Euler iteration and corresponding Sullivan’s basin

The phenomenon of “numerical extraneous roots” of Euler’s iteration has been found. By systematic searching, some polynomials and the corresponding initial values are given, which make the fixed points of Euler’s iteration not the roots of the polynomials. For those repelling extraneous fixed points, the adjoint dynamical types of Sullivan’s basins are also studied. Finally, the fractal pictures are produced.     

Hermite matrix polynomials and second order matrix differential equations

In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y″−xAY′+BY=0. An explicit expression for the Hermite matrix polynomials, the orthogonality property and a Rodrigues’ formula are given.     

ON AN EXTENSION OF ABEL-GONTSCHAROFF＇S EXPANSION FORMULA

We present a constructive generalization of Abel-Gontscharoff＇s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for constructing the basis functions of the interpolants are given.