Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second, third, and fourth kinds

1995 ◽  
Vol 9 (1) ◽  
pp. 39-54 ◽  
Author(s):  
J. C. Mason ◽  
G. H. Elliott
Keyword(s):  
Chebyshev Polynomials ◽  
Minimax Approximation

On the Security of Public-Key Algorithms Based on Chebyshev Polynomials over the Finite Field $Z_N$

2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Author(s):  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials ◽  
Public Key

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

scholarly journals On the growth and zeros of polynomials attached to arithmetic functions

2021 ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Fourier Coefficients ◽  
Arithmetic Functions ◽  
Zeros Of Polynomials ◽  
Simple Complex ◽  
Hook Length ◽  
Moderate Growth ◽  
Growth Properties ◽  
Sum Of Divisors

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

scholarly journals Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Waleed M. Abd-Elhameed ◽  
Youssri H. Youssri
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Connection Coefficients ◽  
Current Article ◽  
In Virtue Of ◽  
Connection Formulas

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

scholarly journals Norm estimates for Chebyshev polynomials, I

2021 ◽  
pp. 105561
Author(s):  
Klaus Schiefermayr ◽  
Maxim Zinchenko
Keyword(s):  
Chebyshev Polynomials ◽  
Norm Estimates

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

scholarly journals Composition Identities of Chebyshev Polynomials, via 2 × 2 Matrix Powers

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 746
Author(s):  
Primo Brandi ◽  
Paolo Emilio Ricci
Keyword(s):  
Chebyshev Polynomials ◽  
Representation Formula ◽  
Complex Matrices ◽  
Lucas Polynomials ◽  
Standard Composition

Starting from a representation formula for 2 × 2 non-singular complex matrices in terms of 2nd kind Chebyshev polynomials, a link is observed between the 1st kind Chebyshev polinomials and traces of matrix powers. Then, the standard composition of matrix powers is used in order to derive composition identities of 2nd and 1st kind Chebyshev polynomials. Before concluding the paper, the possibility to extend this procedure to the multivariate Chebyshev and Lucas polynomials is touched on.

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