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.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals On the (p,q)–Chebyshev Polynomials and Related Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 136 ◽  
Author(s):  
Can Kızılateş ◽  
Naim Tuğlu ◽  
Bayram Çekim
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating functions are given. Then, derivative properties between these first and second kind polynomials, determinant representations, multilateral and multilinear generating functions are derived.

scholarly journals Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 276 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Lee-Chae Jang ◽  
Gwan-Woo Jang
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Bernoulli Polynomials ◽  
Series Expansions ◽  
Linear Combinations ◽  
Lucas Polynomials ◽  
Finite Products ◽  
Derive Fourier Series

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 26 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jongkyum Kwon
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Lucas Polynomials ◽  
Finite Products

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric functions 2F1 and these representations are obtained by explicit computations.

scholarly journals A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 782 ◽  
Author(s):  
Feng Qi ◽  
Can Kızılateş ◽  
Wei-Shih Du
Keyword(s):  
Chebyshev Polynomials ◽  
Closed Formula ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials

2018 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jongkyum Kwon
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Lucas Polynomials ◽  
Finite Products

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions 2F1 and these representations are obtained by explicit computations.

THE ROLE OF POLYETHYLENIMINE IN ENHANCING PERFORMANCE OF GLUTAMATE BIOSENSORS

2018 ◽  
Vol 8 (3) ◽  
pp. 36-41
Author(s):  
Diep Do Thi Hong ◽  
Duong Le Phuoc ◽  
Hoai Nguyen Thi ◽  
Serra Pier Andrea ◽  
Rocchitta Gaia
Keyword(s):  
First Generation ◽  
Good Reproducibility ◽  
Complex Matrices ◽  
Long Term Stability ◽  
In Vitro Characterization ◽  
Glutamate Oxidase

Background: The first biosensor was constructed more than fifty years ago. It was composed of the biorecognition element and transducer. The first-generation enzyme biosensors play important role in monitoring neurotransmitter and determine small quantities of substances in complex matrices of the samples Glutamate is important biochemicals involved in energetic metabolism and neurotransmission. Therefore, biosensors requires the development a new approach exhibiting high sensibility, good reproducibility and longterm stability. The first-generation enzyme biosensors play important role in monitoring neurotransmitter and determine small quantities of substances in complex matrices of the samples. The aims of this work: To find out which concentration of polyethylenimine (PEI) exhibiting the most high sensibility, good reproducibility and long-term stability. Methods: We designed and developed glutamate biosensor using different concentration of PEI ranging from 0% to 5% at Day 1 and Day 8. Results: After Glutamate biosensors in-vitro characterization, several PEI concentrations, ranging from 0.5% to 1% seem to be the best in terms of VMAX, the KM; while PEI content ranging from 0.5% to 1% resulted stable, PEI 1% displayed an excellent stability. Conclusions: In the result, PEI 1% perfomed high sensibility, good stability and blocking interference. Furthermore, we expect to develop and characterize an implantable biosensor capable of detecting glutamate, glucose in vivo. Key words: Glutamate biosensors, PEi (Polyethylenimine) enhances glutamate oxidase, glutamate oxidase biosensors

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