scholarly journals On Power Sums Involving Lucas Functions Sequences

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Stefano Barbero
Keyword(s):  
Chebyshev Polynomials ◽  
Power Sums ◽  
Polynomial Sequences ◽  
Divisibility Properties

We present some general formulas related to sum of powers, also with alternating sign, involving Lucas functions sequences. In particular, our formulas give a synthesis of various identities involving sum of powers of well-known polynomial sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Chebyshev polynomials. Finally, we point out some interesting divisibility properties between polynomials arising from our results.

scholarly journals An Identity Involving the Integral of the First-Kind Chebyshev Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Xiao Wang ◽  
Jiayuan Hu
Keyword(s):  
Open Problem ◽  
Chebyshev Polynomials ◽  
Power Sums ◽  
Interesting Identity

We used the algebraic manipulations and the properties of Chebyshev polynomials to obtain an interesting identity involving the power sums of the integral of the first-kind Chebyshev polynomials and solved an open problem proposed by Wenpeng Zhang and Tingting Wang.

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

scholarly journals Some Identities Involving Chebyshev Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaoxue Li
Keyword(s):  
Chebyshev Polynomials ◽  
Combinatorial Method ◽  
Sums Of Powers ◽  
Divisibility Properties

The main purpose of this paper is using the combinatorial method and algebraic manipulations to study some sums of powers of Chebyshev polynomials and give several interesting identities. As some applications of these results, we obtained several divisibility properties involving Chebyshev polynomials.

scholarly journals Determinant Representations of Polynomial Sequences of Riordan Type

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-liang Yang ◽  
Sai-nan Zheng
Keyword(s):  
Recurrence Relation ◽  
Characteristic Polynomial ◽  
Chebyshev Polynomials ◽  
General Term ◽  
Polynomial Sequence ◽  
Polynomial Sequences ◽  
Principal Submatrix ◽  
Riordan Array

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

scholarly journals Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Tingting Wang ◽  
Han Zhang
Keyword(s):  
Open Problem ◽  
Chebyshev Polynomials ◽  
Combinatorial Method ◽  
Power Sums

We use the combinatorial method and algebraic manipulations to obtain several interesting identities involving the power sums of the derivative of the first kind Chebyshev polynomials. This solved an open problem proposed by Li (2015).

scholarly journals On the Chebyshev Polynomials and Some of Their Reciprocal Sums

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 704
Author(s):  
Wenpeng Zhang ◽  
Di Han
Keyword(s):  
Chebyshev Polynomials ◽  
Mathematical Induction ◽  
Symmetric Polynomial ◽  
Trigonometric Function ◽  
Polynomial Sequences ◽  
Close Relationship ◽  
Riemann Ζ Function ◽  
Reciprocal Sums

In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann ζ-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.

Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

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