New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers, and Jacobsthal-Lucas numbers

2021 ◽  
Vol 150 ◽  
pp. 111173
Author(s):  
Songül Çelik ◽  
İnan Durukan ◽  
Engin Özkan
Keyword(s):  
Lucas Numbers ◽  
Pell Numbers

Pell Numbers, Pell–Lucas Numbers and Modular Group

2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat
Keyword(s):  
Modular Group ◽  
Symmetric Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Pell Numbers ◽  
The Matrix

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

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.

Convolutions of the generalized Pell and Pell-Lucas numbers

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 105-112 ◽  
Author(s):  
Gospava Djordjevic
Keyword(s):  
Lucas Numbers ◽  
Pell Numbers

We consider the convolution of the generalized Pell numbers-P(s) n,m and the convolution of the generalized Pell-Lucas numbers-Q(s) n,m. For s = 0, the sequence P(0) n,m represents the generalized Pell numbers Pn;m, and the sequence Q(0) n,m represents the generalized Pell-Lucas numbers Qn,m ([1], [2]). For m = 2 and s = 0, the numbers P(0) n,2 and Q(0) n,2 are Pell and Pell-Lucas numbers, respectively.

Pell Walks

2013 ◽  
Vol 97 (538) ◽  
pp. 27-35 ◽  
Author(s):  
Thomas Koshy
Keyword(s):  
Lattice Paths ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Fertile Ground ◽  
Fibonacci And Lucas Numbers ◽  
Image Position

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:where n ≥ 3. They can also be defined by Binet-like formulas:

scholarly journals Properties of hyperbolic generalized Pell numbers

2020 ◽  
Vol 26 (4) ◽  
pp. 136-153
Author(s):  
Yüksel Soykan ◽  
◽  
Melih Göcen ◽  
Keyword(s):  
Clifford Algebra ◽  
Generating Functions ◽  
Hyperbolic Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, we introduce the generalized hyperbolic Pell numbers over the bidimensional Clifford algebra of hyperbolic numbers. As special cases, we deal with hyperbolic Pell and hyperbolic Pell–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 to these sequences.

Divisibility properties for Fibonacci and related numbers

2013 ◽  
Vol 97 (540) ◽  
pp. 461-464
Author(s):  
Jawad Sadek ◽  
Russell Euler
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Unified Approach ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Image Position ◽  
Divisibility Properties

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.For non-negative integers n, the recurrence relation defined bywith initial conditionscan be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

scholarly journals ORDINARY GENERATING FUNCTIONS OF BINARY PRODUCTS OF (p,q)-MODIFIED PELL NUMBERS AND k-NUMBERS AT POSITIVE AND NEGATIVE INDICES

2020 ◽  
Vol 20 (3) ◽  
pp. 627-648
Author(s):  
NABIHA SABA ◽  
ALI BOUSSAYOUD
Keyword(s):  
Generating Functions ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Balancing Numbers ◽  
Symmetric Properties

In this paper, we introduce a operator in order to derive some new symmetric properties of (p,q)-modified Pell numbers and we give some new generating functions of the products of (p,q)-modified Pell numbers with k-Fibonacci and k-Lucas numbers, k-Pell and k-Pell Lucas numbers, k-Jacobsthal and k-Jacobsthal Lucas numbers at positive and negative indices. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-modified Pell numbers with k-balancing and k-Lucas-balancing numbers.

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 Gaussian Pell and Gaussian Pell-Lucas quaternions

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1609-1617
Author(s):  
Hasan Arslan
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Summation Formulas

The main aim of this work is to introduce the Gaussian Pell quaternion QGpn and Gaussian Pell-Lucas quaternion QGqn, where the components of QGpn and QGqn are Pell numbers pn and Pell-Lucas numbers qn, respectively. Firstly, we obtain the recurrence relations and Binet formulas for QGpn and QGqn. We use Binet formulas to prove Cassini?s identity for these quaternions. Furthermore, we give some basic identities for QGpn and QGqn such as some summation formulas, the terms with negative indices and the generating functions for these complex quaternions.

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