Two Kinds of Numbers and Their Applications

C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.     

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.     

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.     

Sample Numbers and Optimal Lagrange Interpolation of Sobolev Spaces Wr1*

This paper investigates the optimal recovery of Sobolev spaces W^r₁[?1, 1], r ∈ N in the space L1[?1, 1]. They obtain the values of the sampling numbers of W^r₁[?1, 1] in L₁[?1, 1] and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms. Meanwhile, they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.     

EXACT EVALUATIONS OF FINITE TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli’s numbers play a role similar to that of Euler’s in the old ones. Our technique differs from that of Byrne-Smith (1997) and Berndt-Yeap (2002).     

APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS AND CONVOLUTION INTEGRALS USING LEGENDRE POLYNOMIALS

It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard deviation as a measure of the accuracy of the approximation and the CPU time as a measure of the speed, we find that for reasonable accuracy Legendre polynomials are more efficient. '     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> Extemal Polynomials Associated with Generalized Jacobi Weights

Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.     

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respectto which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zerofree interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu-tion that a Markov system, under an additional assumption, is dense if and only if the maxi-mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.     

ON THE RATE OF WEIGHTED Lp CONVERGENCE BY THE GRUENWALD INTERPOLATORY OPERATORS

This paper gives the weighted Lp convergence rate estimations of the Gruenwald interpolatory polynomials based on the zeros of Chebyshev polymomials of the first kind,and proves that the order of the estimations is optimal for p≥1.     

ON THE RATE OF WEIGHTED L_p CONVERGENCE BY THE GRNWALD INTERPOLATORY OPERATORS

Abstract This paper gives the weighetd L_p convergence rate estimations of the Grunwald interpolatory polynomi-als based on the zeros of Chebyshev polynomials of the first kind, and proves that the order of the estima-tions is optimal for p≥1.     

一些包含Chebyshev多项式和Stirling数的恒等式

利用初等方法研究Chebyshev多项式的性质,建立了广义第二类Chebyshev多项式的一个显明公式,并得到了一些包含第一类Chebyshev多项式,第一类Stirling数和Lucas数的恒等式.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

基于两类切比雪夫多项式的循环矩阵行列式的计算

利用多项式因式分解的逆变换,结合循环矩阵和切比雪夫多项式的特殊结构,首先研究第三类和第四类切比雪夫多项式的通项公式,并给出第三类、第四类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式的显式表达式,最后给出算法实施步骤.     

Adomian decomposition method with orthogonal polynomials: Legendre polynomials

This paper illustrates the using of orthogonal polynomials to modify the Adomian decomposition method. The method of employing Legendre polynomials to improve the Adomian decomposition method is presented here and compared to the method of using Chebyshev polynomials. The presented modified Adomian decomposition method is validated through an example and advantage as well as efficiency of this method is verified through investigating and comparing the results. In this paper, it is concluded that both orthogonal polynomials: Chebyshev and Legendre polynomials can be successfully used for the Adomian decomposition method and comparatively the Chebyshev expansion provides the better estimation.     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

Small polynomials with integer coefficients

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of the integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for the integer Chebyshev constant. Moreover, all the bounds mentioned can be found numerically by using various extremal point techniques, such as the weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials. Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.     

A note on Chebyshev polynomials

New families of generating functions and identities concerning the Chebyshev polynomials are derived. It is shown that the proposed method allows the derivation of sum rules involving products of Chebyshev polynomials and addition theorems. The possiblity of extending the results to include gnerating functions involving products of Chebyshev and other polynomials is finally analyzed.

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Sunto Si derivano nuove famiglie di funzioni generatrici e di identità relative ai polinomi di Chebyshev. Si dimostra che il metodo proposto permette la derivazione di regole di somma relative a prodotti di polinomi di Chebyshev e teoremi di addizione. La possibilità di estendere i risultati includendos funzioni generatrici di prodotti di polinomi di Chebyshev ed altri polinomi è infine analizzata.

     

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.     

三对角行列式与Chebyshev多项式

给出了三对角行列式的几种算法,利用三对角行列式证明了两类Chebyshev多项式的几种显式.