scholarly journals Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions

2017 ◽  
Vol 15 (1) ◽  
pp. 1156-1160 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Recursion Formula ◽  
Derivatives Of

Abstract A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.

scholarly journals On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Li
Keyword(s):  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Fibonacci Polynomials ◽  
Relationship Of ◽  
The Relationship ◽  
Derivatives Of

We study the relationship of the Chebyshev polynomials, Fibonacci polynomials, and theirrth derivatives. We get the formulas for therth derivatives of Chebyshev polynomials being represented by Chebyshev polynomials and Fibonacci polynomials. At last, we get several identities about the Fibonacci numbers and Lucas numbers.

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 New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers

2020 ◽  
Vol 12 (2) ◽  
pp. 280-286
Author(s):  
Carlos M. da Fonseca
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Tridiagonal Matrices

AbstractIn this note, we recall several connections between the determinant of some tridiagonal matrices and the orthogonal polynomials allowing the relation between Chebyshev polynomials of second kind and Fibonacci numbers. With basic transformations, we are able to recover some recent results on this matter, bringing them into one place.

Resultants of Chebyshev Polynomials: the First, Second, Third, and Fourth Kinds

2015 ◽  
Vol 58 (2) ◽  
pp. 423-431 ◽  
Author(s):  
Masakazu Yamagishi
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Cyclotomic Polynomials

AbstractWe give an explicit formula for the resultant ofChebyshev polynomials of the ûrst, second, third, and fourth kinds. We also compute the resultant of modiûed cyclotomic polynomials.

scholarly journals Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

2021 ◽  
Vol 5 (4) ◽  
pp. 165
Author(s):  
Mohamed Abdelhakem ◽  
Toqa Alaa-Eldeen ◽  
Dumitru Baleanu ◽  
Maryam G. Alshehri ◽  
Mamdouh El-Kady
Keyword(s):  
Error Analysis ◽  
Galerkin Method ◽  
Chebyshev Polynomials ◽  
Real Life ◽  
Operational Matrices ◽  
Integer Order ◽  
First Derivative ◽  
The Core ◽  
Life Problems ◽  
Derivatives Of

An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

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