scholarly journals A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials

2020 ◽  
pp. 42-42
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Recurrence Formula ◽  
Binomial Coefficients ◽  
Combinatorial Identity ◽  
Lucas Numbers ◽  
Fundamental Properties ◽  
Special Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers are given.

From golden-ratio equalities to Fibonacci and Lucas identities

2013 ◽  
Vol 97 (539) ◽  
pp. 234-241
Author(s):  
Martin Griffiths
Keyword(s):  
High School Students ◽  
Generating Functions ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
School Students ◽  
Simple Method ◽  
Matrix Methods ◽  
Binet’S Formula

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals On Dual Hyperbolic Generalized Fibonacci Numbers

2019 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

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 Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

scholarly journals Some properties of k-generalized Fibonacci numbers

2021 ◽  
Vol 50 ◽  
pp. 73-79
Author(s):  
Nazmiye Yılmaz ◽  
Ali Aydoğdu ◽  
Engin Özkan
Keyword(s):  
Fibonacci Numbers ◽  
Binomial Coefficients ◽  
Generalized Fibonacci Numbers ◽  
New Family

In the present paper, we propose some properties of the new family 𝑘-generalized Fibonacci numbers which related to generalized Fibonacci numbers. Moreover, we give some identities involving binomial coefficients for 𝑘-generalized Fibonacci numbers.

scholarly journals Construction of a new class of generating functions of binary products of some special numbers and polynomials

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1001-1013
Author(s):  
Souhila Boughaba ◽  
Ali Boussayoud ◽  
Serkan Araci ◽  
Mohamed Kerada ◽  
Mehmet Acikgoz
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pell Numbers ◽  
New Class ◽  
Special Numbers And Polynomials ◽  
Symmetric Properties

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

scholarly journals GENERATING FUNCTIONS OF EVEN AND ODD GAUSSIAN NUMBERS AND POLYNOMIALS

2021 ◽  
Vol 21 (1) ◽  
pp. 125-144
Author(s):  
NABIHA SABA ◽  
ALI BOUSSAYOUD ◽  
MOHAMED KERADA
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
New Class ◽  
Gaussian Polynomials

In this study, we introduce a new class of generating functions of odd and even Gaussian (p,q)-Fibonacci numbers, Gaussian (p,q)-Lucas numbers, Gaussian (p,q)-Pell numbers, Gaussian (p,q)-Pell Lucas numbers, Gaussian Jacobsthal numbers and Gaussian Jacobsthal Lucas numbers and we will recover the new generating functions of some Gaussian polynomials at odd and even terms. The technique used her is based on the theory of the so called symmetric functions.

scholarly journals General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function

2021 ◽  
Vol 55 (2) ◽  
pp. 115-123
Author(s):  
R. Frontczak ◽  
T. Goy
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The purpose of this paper is to present closed forms for various types of infinite seriesinvolving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.To prove our results, we will apply some conventional arguments and combine the Binet formulasfor these sequences with generating functions involving the Riemann zeta function and some known series evaluations.Among the results derived in this paper, we will establish that $\displaystyle\sum_{k=1}^\infty (\zeta(2k+1)-1) F_{2k} = \frac{1}{2},\quad\sum_{k=1}^\infty (\zeta(2k+1)-1) \frac{L_{2k+1}}{2k+1} = 1-\gamma,$ where $\gamma$ is the familiar Euler-Mascheroni constant.

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