scholarly journals Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 334 ◽  
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Fibonacci Polynomials ◽  
Recursive Sequence ◽  
Combinatorial Methods

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.

scholarly journals A New Identity Involving Balancing Polynomials and Balancing Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1141 ◽  
Author(s):  
Yuanyuan Meng
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Second Order ◽  
Balancing Numbers ◽  
Recursive Sequence ◽  
Combinatorial Methods

In this paper, a second-order nonlinear recursive sequence M ( h , i ) is studied. By using this sequence, the properties of the power series, and the combinatorial methods, some interesting symmetry identities of the structural properties of balancing numbers and balancing polynomials are deduced.

scholarly journals Some Identities Involving the Fubini Polynomials and Euler Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 300 ◽  
Author(s):  
Guohui Chen ◽  
Li Chen
Keyword(s):  
Power Series ◽  
Second Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Non Linear ◽  
Combinatorial Methods

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

scholarly journals Some New Identities of Second Order Linear Recurrence Sequences

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1496 ◽  
Author(s):  
Yanyan Liu ◽  
Xingxing Lv
Keyword(s):  
Power Series ◽  
Second Order ◽  
Linear Recurrence ◽  
Combinatorial Method ◽  
Order Linear ◽  
Computational Problem ◽  
Characteristic Roots ◽  
Recursive Sequence ◽  
Recurrence Sequences

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

scholarly journals A New Identity Involving the Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 244 ◽  
Author(s):  
Yixue Zhang ◽  
Zhuoyu Chen
Keyword(s):  
Chebyshev Polynomials ◽  
Second Order ◽  
Simple Problem ◽  
Computational Problem ◽  
Non Linear ◽  
Recursive Sequence ◽  
Interesting Identity ◽  
Combinatorial Methods

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible to simplify a class of complex computations involving the second type Chebyshev polynomials into a very simple problem. Finally, we give a new and interesting identity for it.

scholarly journals On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0 kW3 k and Σn k=1 kW3− k

2020 ◽  
pp. 37-52 ◽  
Author(s):  
Yuksel Soykan
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas Σn k=0 kW3 k and Σn k=1 kW3-k for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

scholarly journals Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

2021 ◽  
pp. 26-37
Author(s):  
Olumide O. Olaiya ◽  
Rasaq A. Azeez ◽  
Mark I. Modebei
Keyword(s):  
Power Series ◽  
Method Of Lines ◽  
Second Order ◽  
Basis Functions ◽  
Direct Solution ◽  
Block Method ◽  
Partial Differential ◽  
Integration Schemes ◽  
The Method Of Lines ◽  
Grid Points

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

scholarly journals Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants

2017 ◽  
Author(s):  
Feng Qi ◽  
Jing-Lin Wang ◽  
Bai-Ni Guo
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Closed Expression

In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

scholarly journals Ups and Downs of Water Photodecolorization by Nanocomposite Polymer Nanofibers

Nanomaterials ◽  
2019 ◽  
Vol 9 (2) ◽  
pp. 250 ◽  
Author(s):  
Shahin Homaeigohar ◽  
Niharika Krishna Botcha ◽  
Eman. S. Zarie ◽  
Mady Elbahri
Keyword(s):  
Structural Properties ◽  
Thermomechanical Properties ◽  
Second Order ◽  
Point Of View ◽  
Order Kinetic ◽  
Dye Molecules ◽  
Nanocomposite Polymer ◽  
Nanocomposite System ◽  
Order Kinetic Model ◽  
Pros And Cons

Given the exponentially expanding water pollution causing water scarcity, there is an urgent need for operative nanotechnological systems that can purify water, with insignificant energy consumption, and rapidly. Here, we introduce a nanocomposite system based on TiO2 nanoparticles (NPs) and PES nanofibers (NFs) that can adsorb and then photodecompose organic water pollutants such as dye molecules. We evaluate pros and cons of this system with respect to its purification efficiency and structural properties that can be impacted by the photocatalytic activity of the nanofillers. While the material is superhydrophilic and able to remove 95% methylene blue (MB) from water via adsorption/photodecomposition, its thermomechanical properties decline upon UV irradiation. However, these properties still remain at the level of the neat NFs. The removal behavior is modeled by the first- and second-order kinetic models from the kinetic point of view. The nanocomposite NFs’ removal behavior complies much better with the second-order kinetic model. Overall, such feedbacks implied that the nanocomposite can be effectively applied for water treatment and the structural properties are still as reliable as those of the neat counterpart.

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