scholarly journals Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2002
Author(s):  
Necdet Batir ◽  
Anthony Sofo
Keyword(s):  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Lucas Numbers ◽  
Finite Sums ◽  
Fibonacci And Lucas Numbers ◽  
Finite Sum

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

scholarly journals Some Finite Sums Involving Generalized Fibonacci and Lucas Numbers

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
E. Kılıç ◽  
N. Ömür ◽  
Y. T. Ulutaş
Keyword(s):  
Lucas Numbers ◽  
Alternating Sums ◽  
Finite Sums ◽  
Fibonacci And Lucas Numbers

By considering Melham's sums (Melham, 2004), we compute various more general nonalternating sums, alternating sums, and sums that alternate according to involving the generalized Fibonacci and Lucas numbers.

scholarly journals Triple sums including Fibonacci numbers with three binomial coefficients

2021 ◽  
Vol 27 (1) ◽  
pp. 188-197
Author(s):  
Funda Taşdemir ◽  
Keyword(s):  
Linear Combination ◽  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

In this paper, we consider some triple sums that involve Fibonacci numbers with three binomial coefficients. We chose the indices of Fibonacci numbers as linear combination of the summation indices. Moreover, various types of alternating analogues of them whose powers depend on the index or indices are computed. These sums are evaluated in nice multiplication forms in terms of Fibonacci and Lucas numbers.

scholarly journals Sums of products of generalized Fibonacci and Lucas numbers

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Recent Work ◽  
Recurrence Relation ◽  
Fixed Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Lucas Numbers ◽  
Alternating Sums ◽  
Explicit Formulae ◽  
Fibonacci And Lucas Numbers ◽  
Sums Of Products

AbstractIn this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.

Finite Mixed Sums wih Harmonic Terms

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Anthony Sofo
Keyword(s):  
Higher Order ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Euler Sums ◽  
Finite Sums ◽  
Sums Of Products

AbstractIn the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

scholarly journals Riordan matrices and sums of harmonic numbers

2011 ◽  
Vol 5 (2) ◽  
pp. 176-200 ◽  
Author(s):  
Emanuele Munarini
Keyword(s):  
Stirling Numbers ◽  
Catalan Numbers ◽  
Subject Classification ◽  
Harmonic Numbers ◽  
Lucas Numbers ◽  
Primary 05A15 ◽  
Secondary 05A10 ◽  
General Identity ◽  
2000 Mathematics Subject Classification ◽  
Fibonacci And Lucas Numbers

We obtain a general identity involving the row-sums of a Riordan matrix and the harmonic numbers. From this identity, we deduce several particular identities involving numbers of combinatorial interest, such as generalized Fibonacci and Lucas numbers, Catalan numbers, binomial and trinomial coefficients, Stirling numbers. 2000 Mathematics Subject Classification: Primary: 05A15; Secondary: 05A10.

scholarly journals Combinatorial sums associated with balancing and Lucas-balancing polynomials

2020 ◽  
Vol Accepted manuscript ◽  
Author(s):  
Frontczak Robert ◽  
Goy Taras
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Combinatorial Sums ◽  
Fibonacci And Lucas Numbers

The aim of the paper is to use some identities involving binomial coefficients to derive new combinatorial identities for balancing and Lucasbalancing polynomials. Evaluating these identities at specific points, we can also establish some combinatorial expressions for Fibonacci and Lucas numbers.

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.

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