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Open AccessArticle

New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences

by Bahar Demirtürk

Mathematics 2025, 13(14), 2321; https://doi.org/10.3390/math13142321 - 21 Jul 2025

Viewed by 511

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities

A_{n}^{2} - A_{n + 1} A_{n - 1} = k^{- 2 n}

A_{n}^{2} - A_{n} A_{n - 1} + \frac{1}{k^{2}} A_{n - 1}^{2} = k^{- 2 n}

, and

B_{n}^{2} - B_{n} B_{n - 1} + \frac{1}{k^{2}} B_{n - 1}^{2} = - (k^{2} - 4) k^{- 2 n}

, and some generalizations such as

B_{n + m}^{2} - (k^{2} - 4) A_{n - t} B_{n + m} A_{t + m} - (k^{2} - 4) k^{2 t - 2 n} A_{t + m}^{2} = k^{- 2 m - 2 t} B_{n - t}^{2}

A_{t + m}^{2} - B_{t - n} A_{n + m} A_{t + m} + k^{2 n - 2 t} A_{n + m}^{2} = k^{- 2 n - 2 m} A_{t - n}^{2}

, and more were derived, where

m, n, t \in ℤ

and

t \neq n

. In addition to this, the solution pairs of the algebraic equations

x^{2} - B_{p} x y + k^{- 2 p} y^{2} = k^{- 2 q} A_{p}^{2}

x^{2} - (k^{2} - 4) A_{p} x y - (k^{2} - 4) k^{- 2 p} y^{2} = k^{- 2 q} B_{p}^{2}

, and

x^{2} - B_{p} x y + k^{- 2 p} y^{2} = - (k^{2} - 4) k^{- 2 q} A_{p}^{2}

are presented, where

A_{p}

and

B_{p}

are k-Oresme and k-Oresme–Lucas numbers, respectively. Full article

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

13 pages, 269 KB

Open AccessArticle

Combinatorial Analysis of k-Oresme and k-Oresme–Lucas Sequences

by Bahar Demirtürk

Symmetry 2025, 17(5), 697; https://doi.org/10.3390/sym17050697 - 2 May 2025

Cited by 1 | Viewed by 639

Abstract

In this study, firstly the definitions and basic algebraic properties of k-Oresme and k-Oresme–Lucas sequences are given. Then, various summation formulae are derived with the help of the first and second derivatives of two polynomials with k-Oresme and k-Oresme–Lucas number coefficients. The main aim of this study is to establish the relations between the generalized Fibonacci and generalized Lucas sequences and the k-Oresme and k-Oresme–Lucas sequences, respectively. These connections allow us to obtain different combinatorial identities of these sequences using the characteristic equation of the k-Oresme and k-Oresme–Lucas sequences. In this way, the discovered combinatorial identities reveal the arithmetic and structural symmetries in the sequences, through the regularities and recurring patterns observed in the algebraic structures of the considered number sequences. The results obtained in this study enable the development of new symmetric approaches in areas such as numerical analysis, cryptography and optimization algorithms, and the algebraic relations derived in this study can contribute to the solution of different problems in disciplines such as mathematical modelling and theoretical physics. Full article

(This article belongs to the Section Mathematics)

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 1629

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)

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