scholarly journals Incomplete Bivariate Fibonacci and Lucas -Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Dursun Tasci ◽  
Mirac Cetin Firengiz ◽  
Naim Tuglu
Keyword(s):  
Generating Function ◽  
Lucas Numbers ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

We define the incomplete bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

BASIS FOR SYNTHESIS OF SPIRAL LATTICE QUASICRYSTALS

1989 ◽  
Vol 03 (14) ◽  
pp. 1071-1085 ◽  
Author(s):  
L. A. BURSILL ◽  
GEORGE RYAN ◽  
XUDONG FAN ◽  
J. L. ROUSE ◽  
JULIN PENG ◽  
...  
Keyword(s):  
Experimental Error ◽  
Theoretical Models ◽  
Cell Number ◽  
Close Packing ◽  
Helianthus Tuberosus ◽  
Lucas Numbers ◽  
Average Growth ◽  
Left And Right ◽  
Seed Cell ◽  
Fibonacci And Lucas Numbers

Observations of the sunflower Helianthus tuberosus reveal the occurrence of both Fibonacci and Lucas numbers of visible spirals (parastichies). This species is multi-headed, allowing a quantitative study of the relative abundance of these two types of phyllotaxis. The florets follow a spiral arrangement. It is remarkable that the Lucas series occurred, almost invariably, in the first-flowering heads of individual plants. The occurrence of left-and right-handed chirality was found to be random, within experimental error, using an appropriate chirality convention. Quantitative crystallographic studies allow the average growth law to be derived (r = alτ−1; θ = 2πl/(τ + 1), where a is a constant, l is the seed cell number and τ is the golden mean [Formula: see text]). They also reveal departures from classical theoretical models of phyllotaxis, taking the form of persistent oscillations in both divergence angle and radius. The experimental results are discussed in terms of a new theoretical model for the close-packing of growing discs. Finally, a basis for synthesis of (inorganic) spiral lattice structures is proposed.

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

scholarly journals On the k -Lucas Numbers and Lucas Polynomials

2017 ◽  
Vol 5 (4) ◽  
pp. 121-125 ◽  
Author(s):  
Ali Boussayoud ◽  
Mohamed Kerada ◽  
Nesrine Harrouche
Keyword(s):  
Lucas Numbers ◽  
Lucas Polynomials

scholarly journals Identities on generalized Fibonacci and Lucas numbers

2020 ◽  
Vol 26 (3) ◽  
pp. 189-202
Author(s):  
K. M. Nagaraja ◽  
◽  
P. Dhanya ◽  
Keyword(s):  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

2015 ◽  
Vol 168 (2) ◽  
pp. 161-186
Author(s):  
Hajime Kaneko ◽  
Takeshi Kurosawa ◽  
Yohei Tachiya ◽  
Taka-aki Tanaka
Keyword(s):  
Lucas Numbers ◽  
Infinite Products ◽  
Algebraic Dependence ◽  
Fibonacci And Lucas Numbers
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