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27 pages, 370 KB

Open AccessArticle

New Results for Certain Jacobsthal-Type Polynomials

by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori and Amr Kamel Amin

Mathematics 2025, 13(5), 715; https://doi.org/10.3390/math13050715 - 22 Feb 2025

Viewed by 695

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing [...] Read more.

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications, 2nd 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 813

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 2 | Viewed by 822

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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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 883

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

12 pages, 534 KB

Open AccessArticle

The Roots of the Reliability Polynomials of Circular Consecutive-k-out-of-n:F Systems

by Marilena Jianu, Leonard Dăuş, Vlad-Florin Drăgoi and Valeriu Beiu

Mathematics 2023, 11(20), 4252; https://doi.org/10.3390/math11204252 - 11 Oct 2023

Cited by 1 | Viewed by 1363

Abstract

The zeros of the reliability polynomials of circular consecutive-k-out-of-n:F systems are studied. We prove that, for any fixed

k \geq 2

, the set of the roots of all the reliability polynomials (for all

n \geq k

) is [...] Read more.

The zeros of the reliability polynomials of circular consecutive-k-out-of-n:F systems are studied. We prove that, for any fixed

k \geq 2

, the set of the roots of all the reliability polynomials (for all

n \geq k

) is unbounded in the complex plane. In the particular case

k = 2

, we show that all the nonzero roots are real, distinct numbers and find the closure of the set of roots. For every

n \geq k

, the expressions of the minimum root and the maximum root are given, both for circular as well as for linear systems. Full article

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13 pages, 311 KB

Open AccessArticle

p-Numerical Semigroups of Generalized Fibonacci Triples

by Takao Komatsu, Shanta Laishram and Pooja Punyani

Symmetry 2023, 15(4), 852; https://doi.org/10.3390/sym15040852 - 3 Apr 2023

Cited by 9 | Viewed by 1778

Abstract

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup

S_{p}

is defined as the set of integers whose nonnegative integral linear combinations [...] Read more.

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup

S_{p}

is defined as the set of integers whose nonnegative integral linear combinations of given positive integers

a_{1}, a_{2}, \dots, a_{k}

are expressed in more than p ways. When

p = 0

,

S_{0}

with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties. Full article

(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)

18 pages, 372 KB

Open AccessArticle

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

by Takao Komatsu and Claudio Pita-Ruiz

Axioms 2023, 12(2), 98; https://doi.org/10.3390/axioms12020098 - 17 Jan 2023

Cited by 15 | Viewed by 2244

Abstract

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For

p = 0

, the problem is reduced [...] Read more.

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For

p = 0

, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for

p = 0

and by Komatsu and Ying for the general non-negative integer p. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

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11 pages, 282 KB

Open AccessArticle

Distance Fibonacci Polynomials by Graph Methods

by Dominik Strzałka, Sławomir Wolski and Andrzej Włoch

Symmetry 2021, 13(11), 2075; https://doi.org/10.3390/sym13112075 - 3 Nov 2021

Cited by 4 | Viewed by 1978

Abstract

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, [...] Read more.

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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