scholarly journals Alternative proofs of some formulas for two tridiagonal determinants

2018 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Feng Qi ◽  
Ai-Qi Liu
Keyword(s):  
Chebyshev Polynomials ◽  
Tridiagonal Matrix ◽  
Fibonacci Polynomials ◽  
Detailed Proof ◽  
Delannoy Numbers ◽  
Appl Math ◽  
Math Phys

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals The inverse of a triangular matrix and several identities of the Catalan numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 518-541 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular ◽  
Derivatives Of

In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, conclude the inverse of an integer, unit, and lower triangular matrix, derive an inversion theorem, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials respectively.

scholarly journals Bidiagonalization of (k, k + 1)-tridiagonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 20-26 ◽  
Author(s):  
S. Takahira ◽  
T. Sogabe ◽  
T.S. Usuda
Keyword(s):  
Tridiagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonalization ◽  
Tridiagonal Matrices ◽  
The Matrix ◽  
Appl Math ◽  
Diagonalization Algorithm

Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Jaroslav Seibert ◽  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Tridiagonal Matrix ◽  
Lucas Numbers ◽  
Tridiagonal Matrices ◽  
Fibonacci And Lucas Numbers

AbstractThe aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].

Remarks on Chebyshev polynomials, Fibonacci polynomials and Kauffman bracket skein modules

2018 ◽  
Vol 27 (07) ◽  
pp. 1841007
Author(s):  
Robert Owczarek
Keyword(s):  
Chebyshev Polynomials ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Matrix Representations ◽  
Fibonacci Polynomials ◽  
Spin Structures ◽  
Skein Modules ◽  
Kauffman Bracket ◽  
Special Cases ◽  
Counting Solutions

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

scholarly journals Determinantal representations for the number of subsequences without isolated odd terms

2021 ◽  
Vol 27 (4) ◽  
pp. 116-121
Author(s):  
Milica Anđelic ◽  
◽  
Carlos M. da Fonseca ◽  
◽  
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Short Note ◽  
Tridiagonal Matrix ◽  
Hessenberg Matrix

In this short note we propose two determinantal representations for the number of subsequences without isolated odd terms are presented. One is based on a tridiagonal matrix and other on a Hessenberg matrix. We also establish a new explicit formula for the terms of this sequence based on Chebyshev polynomials of the second kind.

scholarly journals Construction of a new class of generating functions of binary products of some special numbers and polynomials

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1001-1013
Author(s):  
Souhila Boughaba ◽  
Ali Boussayoud ◽  
Serkan Araci ◽  
Mohamed Kerada ◽  
Mehmet Acikgoz
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pell Numbers ◽  
New Class ◽  
Special Numbers And Polynomials ◽  
Symmetric Properties

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

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.

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