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.

scholarly journals Three-fold symmetric Hahn-classical multiple orthogonal polynomials

2019 ◽  
Vol 18 (02) ◽  
pp. 271-332 ◽  
Author(s):  
Ana F. Loureiro ◽  
Walter Van Assche
Keyword(s):  
Asymptotic Behavior ◽  
Orthogonal Polynomials ◽  
Symmetric Polynomial ◽  
Recurrence Coefficients ◽  
Absolute Value ◽  
Polynomial Sequences ◽  
Full Characterization ◽  
Classical Polynomials ◽  
Multiple Orthogonal Polynomials

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

scholarly journals Some Identities Involving the Reciprocal Sums of One-Kind Chebyshev Polynomials

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Yuankui Ma ◽  
Xingxing Lv
Keyword(s):  
Chebyshev Polynomials ◽  
Computational Method ◽  
Computational Problem ◽  
Analytic Methods ◽  
Reciprocal Sums

We use the elementary and analytic methods and the properties of Chebyshev polynomials to study the computational problem of the reciprocal sums of one-kind Chebyshev polynomials and give several interesting identities for them. At the same time, we also give a general computational method for this kind of reciprocal sums.

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 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 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 On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Lan Zhang ◽  
Wenpeng Zhang
Keyword(s):  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Mathematical Induction ◽  
Lucas Numbers ◽  
Sums Of Powers

The main purpose of this paper is using mathematical induction and the Girard and Waring formula to study a problem involving the sums of powers of the Chebyshev polynomials and prove some divisible properties. We obtained two interesting congruence results involving Fibonacci numbers and Lucas numbers as some applications of our theorem.

scholarly journals The development of student activity sheet on trigonometric material based on local culture

2021 ◽  
Vol 4 (1) ◽  
pp. 67-78
Author(s):  
Meryani Lakapu ◽  
Wilfridus Beda Nuba Dosinaeng ◽  
Samuel Igo Leton
Keyword(s):  
Development Stage ◽  
Trigonometric Function ◽  
Local Culture ◽  
Design Stage ◽  
Learning Tools ◽  
Student Activity ◽  
Analysis Task ◽  
Close Relationship ◽  
Good Learning ◽  
Research Type

Mathematics and culture are two different things, but they have a very close relationship in everyday life included in learning activities. Therefore, this research aims to describes the process and results of developing student activity sheets based on local culture on Simple Trigonometric Function Graphs. This research type is research and development. The product developed in this research is a student activity sheet based on local culture on Simple Trigonometric Function Graphics. Student activity sheets are developed based on a modified 4-D development model, which consists of defining, designing, and developing. At the definition stage; conducted a preliminary analysis, student analysis, material analysis, task analysis and specification of learning objectives. At the design stage; The preparation of student activity sheets based on local culture is carried out, selecting the format and then doing the initial design. At the development stage; The design results are validated by the expert and then revised according to the expert's notes. From the results of the research and data analysis conducted by researchers, it was found that the student activity sheets developed had met the criteria for good learning tools because they were declared valid, practical and effective.

Minding the Close Relationship

1999 ◽  
Author(s):  
John H. Harvey ◽  
Julia Omarzu
Keyword(s):  
Close Relationship

The Solution Strategy as an Indicator of the Developmental Stage of Preschool Children’s Mental-Rotation Ability

2010 ◽  
Vol 31 (2) ◽  
pp. 95-100 ◽  
Author(s):  
Claudia Quaiser-Pohl ◽  
Anna M. Rohe ◽  
Tobias Amberger
Keyword(s):  
Preschool Children ◽  
Mental Rotation ◽  
Latent Class ◽  
Preschool Age ◽  
Stage Model ◽  
Solution Strategy ◽  
Thinking Aloud ◽  
Transition Analysis ◽  
Solution Strategies ◽  
Close Relationship

The solution strategies of preschool children solving mental-rotation tasks were analyzed in two studies. In the first study n = 111 preschool children had to demonstrate their solution strategy in the Picture Rotation Test (PRT) items by thinking aloud; seven different strategies were identified. In the second study these strategies were confirmed by latent class analysis (LCA) with the PRT data of n = 565 preschool children. In addition, a close relationship was found between the solution strategy and children’s age. Results point to a stage model for the development of mental-rotation ability as measured by the PRT, going from inappropriate strategies like guessing or comparing details, to semiappropriate approaches like choosing the stimulus with the smallest angle discrepancy, to a holistic or analytic strategy. A latent transition analysis (LTA) revealed that the ability to mentally rotate objects can be influenced by training in the preschool age.

Speeded Paper-Pencil Sustained Attention and Mental Speed Tests

2008 ◽  
Vol 29 (4) ◽  
pp. 205-216 ◽  
Author(s):  
Stefan Krumm ◽  
Lothar Schmidt-Atzert ◽  
Kurt Michalczyk ◽  
Vanessa Danthiir
Keyword(s):  
Sustained Attention ◽  
Time Pressure ◽  
Factor Model ◽  
Higher Order ◽  
Future Research ◽  
Perceptual Speed ◽  
Speed Factor ◽  
Close Relationship ◽  
Task Processing ◽  
Order Factor

Mental speed (MS) and sustained attention (SA) are theoretically distinct constructs. However, tests of MS are very similar to SA tests that use time pressure as an impeding condition. The performance in such tasks largely relies on the participants’ speed of task processing (i.e., how quickly and correctly one can perform the simple cognitive tasks). The present study examined whether SA and MS are empirically the same or different constructs. To this end, 24 paper-pencil and computerized tests were administered to 199 students. SA turned out to be highly related to MS task classes: substitution and perceptual speed. Furthermore, SA showed a very close relationship with the paper-pencil MS factor. The correlation between SA and computerized speed was considerably lower but still high. In a higher-order general speed factor model, SA had the highest loading on the higher-order factor; the higher-order factor explained 88% of SA variance. It is argued that SA (as operationalized with tests using time pressure as an impeding condition) and MS cannot be differentiated, at the level of broad constructs. Implications for neuropsychological assessment and future research are discussed.

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