On sums of squares of Fibonomial coefficients by q-calculus

2016 ◽  
Vol 09 (03) ◽  
pp. 1650062
Author(s):  
Emrah Kılıç ◽  
Aynur Yalçıner
Keyword(s):  
Closed Form ◽  
Generating Functions ◽  
Sums Of Squares ◽  
Classical Formula ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

We present some new kinds of sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. As proof method, we will follow the method given in [E. Kılıç and H. Prodinger, Closed form evaluation of sums containing squares of Fibonomial coefficients, accepted in Math. Slovaca]. For this, first we translate everything into [Formula: see text]-notation, and then to use generating functions and Rothe’s identity from classical [Formula: see text]-calculus.

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

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.

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 sums related to the numerator of generating functions for the kth power of Fibonacci numbers on sums related to the numerator of generating functions for the kth power of Fibonacci numbers

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Pavel Pražák ◽  
Pavel Trojovský
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Similar Type ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Sums Of Products

AbstractNew results about some sums s n(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $$ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $$ are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.

scholarly journals Some properties of the generalized Fibonacci and Lucas numbers

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2655-2665
Author(s):  
Gospava Djordjevic ◽  
Snezana Djordjevic
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Fibonacci And Lucas Numbers

In this paper we consider the generalized Fibonacci numbers Fn,m and the generalized Lucas numbers Ln,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.

scholarly journals On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1334
Author(s):  
Rifat Battaloglu ◽  
Yilmaz Simsek
Keyword(s):  
Atomic Structure ◽  
Generating Functions ◽  
Functional Equations ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.

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.

Sign in / Sign up

Export Citation Format

Share Document