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 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.

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 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.

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].

Staircase words and Chebyshev polynomials

2010 ◽  
Vol 4 (1) ◽  
pp. 81-95 ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour ◽  
Augustine Munagi ◽  
Helmut Prodinger
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Trigonometric Sums ◽  
Cyclic Words

A word ? = ?1... ?n over the alphabet [k]={1,2,...,k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word ? is said to be staircase-cyclic if it is a staircase word and in addition satisfies |?n??1|?1. We find the explicit generating functions for the number of staircase words and staircase-cyclic words in [k]n, in terms of Chebyshev polynomials of the second kind. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclic words that are not equivalent up to rotation.

scholarly journals Tutte’s invariant approach for Brownian motion reflected in the quadrant

2017 ◽  
Vol 21 ◽  
pp. 220-234 ◽  
Author(s):  
S. Franceschi ◽  
Kilian Raschel
Keyword(s):  
Functional Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Quarter Plane ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Invariant Approach ◽  
Continuous Setting

We consider a Brownian motion with negative drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte’s invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev 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 ◽  
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.

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