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26 pages, 309 KB

Open AccessArticle

Overview of Six Number/Polynomial Sequences Defined by Quadratic Recurrence Relations

by Yujie Kang, Marta Na Chen and Wenchang Chu

Symmetry 2025, 17(5), 714; https://doi.org/10.3390/sym17050714 - 7 May 2025

Cited by 1 | Viewed by 819

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and [...] Read more.

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)

11 pages, 268 KB

Open AccessArticle

A Note on Generalized k-Order F&L Hybrinomials

by Süleyman Aydınyüz and Gül Karadeniz Gözeri

Axioms 2025, 14(1), 41; https://doi.org/10.3390/axioms14010041 - 5 Jan 2025

Cited by 1 | Viewed by 1023

Abstract

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within [...] Read more.

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

16 pages, 307 KB

Open AccessArticle

Horadam–Lucas Cubes

by Elif Tan, Luka Podrug and Vesna Iršič Chenoweth

Axioms 2024, 13(12), 837; https://doi.org/10.3390/axioms13120837 - 28 Nov 2024

Cited by 3 | Viewed by 1123

Abstract

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family [...] Read more.

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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11 pages, 256 KB

Open AccessArticle

A New Approach to k-Oresme and k-Oresme-Lucas Sequences

by Engin Özkan and Hakan Akkuş

Symmetry 2024, 16(11), 1407; https://doi.org/10.3390/sym16111407 - 22 Oct 2024

Cited by 8 | Viewed by 1603

Abstract

In this study, the

k

-Oresme and

k

-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the

k

-Oresme and

k

-Oresme-Lucas sequences are presented. In addition, we give these sequences the [...] Read more.

In this study, the

k

-Oresme and

k

-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the

k

-Oresme and

k

-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the

k

-Oresme and

k

-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

24 pages, 297 KB

Open AccessArticle

Novel Classes on Generating Functions of the Products of (p,q)-Modified Pell Numbers with Several Bivariate Polynomials

by Ali Boussayoud, Salah Boulaaras and Ali Allahem

Mathematics 2024, 12(18), 2902; https://doi.org/10.3390/math12182902 - 18 Sep 2024

Viewed by 1034

Abstract

In this paper, using the symmetrizing operator

δ_{e_{1} e_{2}}^{2 - l}

, we derive new generating functions of the products of

(p, q)

-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and [...] Read more.

In this paper, using the symmetrizing operator

δ_{e_{1} e_{2}}^{2 - l}

, we derive new generating functions of the products of

(p, q)

-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials. Full article

7 pages, 224 KB

Open AccessArticle

Markov Triples with Generalized Pell Numbers

by Julieth F. Ruiz, Jose L. Herrera and Jhon J. Bravo

Mathematics 2024, 12(1), 108; https://doi.org/10.3390/math12010108 - 28 Dec 2023

Viewed by 1913

Abstract

For an integer

k \geq 2

, let

{(P_{n}^{(k)})}_{n}

be the k-generalized Pell sequence which starts with

0, \dots, 0, 1

(k terms), and each term afterwards is given by [...] Read more.

For an integer

k \geq 2

, let

{(P_{n}^{(k)})}_{n}

be the k-generalized Pell sequence which starts with

0, \dots, 0, 1

(k terms), and each term afterwards is given by

P_{n}^{(k)} = 2 P_{n - 1}^{(k)} + P_{n - 2}^{(k)} + \dots + P_{n - k}^{(k)}

. In this paper, we determine all solutions of the Markov equation

x^{2} + y^{2} + z^{2} = 3 x y z

, with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components. Full article

(This article belongs to the Section A: Algebra and Logic)

15 pages, 288 KB

Open AccessArticle

On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers

by Elen Viviani Pereira Spreafico, Paula Catarino and Paulo Vasco

Axioms 2023, 12(11), 1047; https://doi.org/10.3390/axioms12111047 - 11 Nov 2023

Cited by 1 | Viewed by 2076

Abstract

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper [...] Read more.

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

11 pages, 278 KB

Open AccessArticle

Reciprocal Formulae among Pell and Lucas Polynomials

by Mei Bai, Wenchang Chu and Dongwei Guo

Mathematics 2022, 10(15), 2691; https://doi.org/10.3390/math10152691 - 29 Jul 2022

Cited by 7 | Viewed by 1931

Abstract

Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)

18 pages, 328 KB

Open AccessArticle

Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

by Waleed Mohamed Abd-Elhameed, Andreas N. Philippou and Nasr Anwer Zeyada

Mathematics 2022, 10(13), 2342; https://doi.org/10.3390/math10132342 - 4 Jul 2022

Cited by 27 | Viewed by 2372

Abstract

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind

_{2} F_{1} (z)

are included in all connection coefficients for a specific z. [...] Read more.

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind

_{2} F_{1} (z)

are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed. Full article

13 pages, 280 KB

Open AccessArticle

On (k,p)-Fibonacci Numbers

by Natalia Bednarz

Mathematics 2021, 9(7), 727; https://doi.org/10.3390/math9070727 - 28 Mar 2021

Cited by 10 | Viewed by 2697

Abstract

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix [...] Read more.

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too. Full article

10 pages, 236 KB

Open AccessArticle

On the Reciprocal Sums of Products of Balancing and Lucas-Balancing Numbers

by Younseok Choo

Mathematics 2021, 9(4), 350; https://doi.org/10.3390/math9040350 - 10 Feb 2021

Cited by 3 | Viewed by 1825

Abstract

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and [...] Read more.

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and Lucas-balancing numbers. The method of this paper can also be applied to even-indexed and odd-indexed Fibonacci, Lucas, Pell and Pell–Lucas numbers. Full article

(This article belongs to the Section E: Applied Mathematics)

8 pages, 436 KB

Open AccessArticle

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

by Pavel Trojovský

Mathematics 2020, 8(8), 1387; https://doi.org/10.3390/math8081387 - 18 Aug 2020

Cited by 3 | Viewed by 3301

Abstract

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about [...] Read more.

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence

{(T_{n}^{(k)})}_{n \geq 0}

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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