On modified Chebyshev polynomials

In this paper some new properties and applications of modified Chebyshev polynomials and Morgan-Voyce polynomials will be presented. The aim of the paper is to complete the knowledge about all of these types of polynomials.     

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．     

Symmetrized Chebyshev polynomials

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that and are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.

     

Matrix Chebyshev polynomials and continued fractions

In the first part we expose the notion of continued fractions in the matrix case. In this paper we are interested in their connection with matrix orthogonal polynomials.

In the second part matrix continued fractions are used to develop the notion of matrix Chebyshev polynomials. In the case of hermitian coefficients in the recurrence formula, we give the explicit formula for the Stieltjes transform, the support of the orthogonality measure and its density. As a corollary we get the extension of the matrix version of the Blumenthal theorem proved in [J. Approx. Theory 84 (1) (1996) 96].

The third part contains examples of matrix orthogonal polynomials.     

Cardano's formula, square roots, Chebyshev polynomials and radicals

This paper is focused on the adaptation of Cardano's approach to generating the roots of rescaled Vieta-Lucas polynomials and Vieta-Fibonacci functions. The application of the derived equations to generating radical-type identities is also presented.     

Computations with half-range Chebyshev polynomials

An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) [5]. Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev-Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix.     

On modified Chebyshev polynomials

Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321-343].     

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m ⁴ n ^–6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday     

Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let ${\mathbb{F}}_q$ be a field of q elements, where q is a power of an odd prime p. The polynomial $f(y)\in {\mathbb{F}}_q[y]$ defined by$f(y):={(1+\sqrt{y})}^{(q+1)/2}+{(1-\sqrt{y})}^{(q+1)/2}$ has the property that$f(1-y)=\rho (2)f(y),$ where ρ is the quadratic character on ${\mathbb{F}}_q$. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.     

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.     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

包含切比雪夫多项式的循环矩阵行列式的计算

行首加r尾r右循环矩阵和行尾加r首r左循环矩阵是两种特殊类型的矩阵,这篇论文中就是利用多项式因式分解的逆变换这一重要的技巧以及这类循环矩阵漂亮的结构和切比雪夫多项式的特殊的结构,分别讨论了第一类、第二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,从而给出了行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式显式表达式.这些显式表达式与切比雪夫多项式以及参数r有关.这一问题的应用背景主要在循环编码,图像处理等信息理论方面.     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

Extremal properties of certain trigonometric functions and Chebyshev polynomials

For a wide class of symmetric trigonometric polynomials, the minimal deviation property is established. As a corollary, the extremal property is proved for the components of the Chebyshev polynomial mappings corresponding to symmetric algebras A _α.     

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.     

勒让德多项式的性质与契贝谢夫多项式间的关系

主要讨论了著名的勒让德多项式的一些性质,同时得到勒让德多项式与契贝谢夫多项式之间的一些关系